ACTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS ON SPACES OF JACOBI DIAGRAMS. II

Abstract

The automorphism group $\operatorname {Aut}(F_n)$ of the free group $F_n$ acts on a space $A_d(n)$ of Jacobi diagrams of degree d on n oriented arcs. We study the $\operatorname {Aut}(F_n)$-module structure of $A_d(n)$ by using two actions on the associated graded vector space of $A_d(n)$: an action of the general linear group $\operatorname {GL}(n,\mathbb {Z})$ and an action of the graded Lie algebra $\mathrm {gr}(\operatorname {IA}(n))$ of the IA-automorphism group $\operatorname {IA}(n)$ of $F_n$ associated with its lower central series. We extend the action of $\mathrm {gr}(\operatorname {IA}(n))$ to an action of the associated graded Lie algebra of the Andreadakis filtration of the endomorphism monoid of $F_n$. By using this action, we study the $\operatorname {Aut}(F_n)$-module structure of $A_d(n)$. We obtain an indecomposable decomposition of $A_d(n)$ as $\operatorname {Aut}(F_n)$-modules for $n\geq 2d$. Moreover, we obtain the radical filtration of $A_d(n)$ for $n\geq 2d$ and the socle of $A_3(n)$.

1 Introduction

Jacobi diagrams are uni-trivalent graphs, which graphically encode the algebraic structures of Lie algebras and their representations. Jacobi diagrams were introduced for the Kontsevich integral, which is a universal finite type link invariant and unifies all quantum link invariants [2, 18, 15, 19]. The associated graded vector space of finite type link invariants is isomorphic to the space of weight systems, which is the dual to the space of Jacobi diagrams.

Let $\Bbbk $ be a field of characteristic $0$ . We study the $\Bbbk $ -vector space $A(n)$ of Jacobi diagrams on n-component oriented arcs, which is the target space of the Kontsevich integral for string links [8, 3] or bottom tangles [9]. We consider the degree d part $A_d(n)$ of $A(n)$ , where the degree of a Jacobi diagram is determined by half the number of its vertices. The space $A_d(n)$ encodes the universal enveloping algebra $U(\mathfrak {g})$ of any finite-dimensional semisimple Lie algebra $\mathfrak {g}$ . More precisely, the weight system maps $A_d(n)$ to the $\mathfrak {g}$ -invariant part of $U(\mathfrak {g})^{\otimes n}$ .

We consider a filtration for $A_d(n)$ defined by the number of trivalent vertices. The associated graded vector space of $A_d(n)$ is identified via the PBW (Poincaré–Birkhoff–Witt) map [2, 3] with a graded vector space $B_d(n)$ of open Jacobi diagrams of degree d that are colored by elements of an n-dimensional $\Bbbk $ -vector space. For a finite-dimensional semisimple Lie algebra $\mathfrak {g}$ , the weight system maps $B_d(n)$ to the $\mathfrak {g}$ -invariant part of the tensor product $\mathfrak {S}(\mathfrak {g})^{\otimes n}$ of the symmetric algebra $\mathfrak {S}(\mathfrak {g})$ of $\mathfrak {g}$ .

In a previous paper [16], we proved that the vector spaces $A_d(n)$ define a functor $A_d:\mathbf {F}^{\mathrm {op}}\rightarrow \mathbf {fVect}$ from the opposite category $\mathbf {F}^{\mathrm {op}}$ of the category $\mathbf {F}$ of finitely generated free groups to the category $\mathbf {fVect}$ of filtered vector spaces. By functoriality on $\mathbf {F}^{\mathrm {op}}$ , $A_d(n)$ inherits an action of the automorphism group $\operatorname {Aut}(F_n)$ and of the endomorphism monoid $\operatorname {End}(F_n)$ of the free group $F_n$ of rank n. We proved in [16] that the action of $\operatorname {Aut}(F_n)$ on $A_d(n)$ induces an action of the outer automorphism group $\operatorname {Out}(F_n)$ of $F_n$ on $A_d(n)$ and we observed that the $\operatorname {Aut}(F_n)$ -action on $A_d(n)$ induces two actions on $B_d(n)$ : an action of the general linear group $\operatorname {GL}(n;\mathbb {Z})$ and an action of the graded Lie algebra $\mathrm {gr}(\operatorname {IA}(n))$ of the IA-automorphism group $\operatorname {IA}(n)$ of $F_n$ associated with the lower central series. We used these two actions on $B_d(n)$ to study the $\operatorname {Aut}(F_n)$ -module structure of $A_d(n)$ for $d=2$ . However, it is rather difficult to compute the $\mathrm {gr}(\operatorname {IA}(n))$ -action on $B_d(n)$ directly for general d.

The aim of the present paper is to study the $\operatorname {Aut}(F_n)$ -module structure of $A_d(n)$ for general d and especially $d=3$ in detail. We consider the Andreadakis filtration $\mathcal {E}_{\ast }(n)$ of the endomorphism monoid $\operatorname {End}(F_n)$ of $F_n$ . We extend the action of the graded Lie algebra $\mathrm {gr}(\operatorname {IA}(n))$ to an action of the associated graded Lie algebra $\mathrm {gr}(\mathcal {E}_{\ast }(n))$ of the Andreadakis filtration. On the other hand, we construct a graphical version of the $\mathrm {gr}(\mathcal {E}_{\ast }(n))$ -action on $B_d(n)$ . By using this graphical action, we study the $\operatorname {Aut}(F_n)$ -module structure of $A_d(n)$ . We obtain an indecomposable decomposition of $A_d(n)$ as $\operatorname {Aut}(F_n)$ -modules for $n\geq 2d$ . Moreover, we obtain the radical filtration of $A_d(n)$ for $n\geq 2d$ and the socle of $A_3(n)$ .

1.1 Andreadakis filtration of $\operatorname {End}(F_n)$

Let $\Gamma _r:=\Gamma _r(F_n)$ denote the r-th term of the lower central series of the free group $F_n$ . Let $\mathcal {L}_r(n) :=\Gamma _r/\Gamma _{r+1}$ for $r\geq 1$ , and set $H:=\mathcal {L}_1(n)$ . Note that $\mathcal {L}_r(n)$ is the degree r part of the free Lie algebra $\mathcal {L}_{\ast }(n)$ on H.

Let $\operatorname {IA}(n)$ denote the IA-automorphism group of $F_n$ , which is the kernel of the canonical homomorphism $\operatorname {Aut}(F_n)\rightarrow \operatorname {GL}(n;\mathbb {Z})$ .

The Andreadakis filtration $\mathcal {A}_{\ast }(n)$ of $\operatorname {Aut}(F_n)$ [1, 22]

$$ \begin{align*}\operatorname{Aut}(F_n)=\mathcal{A}_0(n)\supset \mathcal{A}_{1}(n)=\operatorname{IA}(n)\supset \mathcal{A}_2(n)\supset \cdots\end{align*} $$

is defined by

$$ \begin{align*}\mathcal{A}_{r}(n)=\ker(\operatorname{Aut}(F_n)\rightarrow \operatorname{Aut}(F_n/\Gamma_{r+1})).\end{align*} $$

For $r\geq 1$ , we have an injective homomorphism

$$ \begin{align*}\tau_r: \operatorname{gr}^r(\mathcal{A}_{\ast}(n))\hookrightarrow \operatorname{Hom}(H,\mathcal{L}_{r+1}(n)),\end{align*} $$

which is called the Johnson homomorphism. By Andreadakis [1] and Kawazumi [17], we have $\operatorname {gr}^1(\operatorname {IA}(n))\cong \operatorname {gr}^1(\mathcal {A}_{\ast }(n))\cong \operatorname {Hom}(H,\mathcal {L}_{2}(n))$ .

We construct the Andreadakis filtration $\mathcal {E}_{\ast }(n)$ of $\operatorname {End}(F_n)$ in a similar way by

$$ \begin{align*}\mathcal{E}_{r}(n)=\ker(\operatorname{End}(F_n)\rightarrow \operatorname{End}(F_n/\Gamma_{r+1})).\end{align*} $$

We define an equivalence relation on the monoid $\mathcal {E}_{r}(n)$ and consider the quotient group $\operatorname {gr}^r(\mathcal {E}_{\ast }(n))$ , which includes $\operatorname {gr}^r(\mathcal {A}_{\ast }(n))$ (see Section 3.3). We also construct the Johnson homomorphism

$$ \begin{align*}\tilde{\tau}_r:\operatorname{gr}^r(\mathcal{E}_{\ast}(n))\xrightarrow{\cong}\operatorname{Hom}(H,\mathcal{L}_{r+1}(n)) \end{align*} $$

of $\operatorname {End}(F_n)$ , which turns out to be an abelian group isomorphism (see Proposition 3.8).

The target group $\operatorname {Hom}(H,\mathcal {L}_{r+1}(n))\cong H^{\ast }\otimes \mathcal {L}_{r+1}(n)$ of the Johnson homomorphism is identified with the degree r part $\operatorname {Der}_r(\mathcal {L}_{\ast }(n))$ of the derivation Lie algebra $\operatorname {Der}(\mathcal {L}_{\ast }(n))$ of the free Lie algebra $\mathcal {L}_{\ast }(n)$ and with the tree module $T_r(n)$ , which we define in Section 3.2. From the above, we have abelian group isomorphisms

$$ \begin{align*}\operatorname{gr}^r(\mathcal{E}_{\ast}(n))\cong H^{\ast}\otimes \mathcal{L}_{r+1}(n)\cong \operatorname{Der}_r(\mathcal{L}_{\ast}(n)) \cong T_r(n). \end{align*} $$

Thus, we have

$$ \begin{align*}\operatorname{gr}^1(\operatorname{IA}(n))\cong \operatorname{gr}^1(\mathcal{E}_{\ast}(n))\cong H^{\ast}\otimes \mathcal{L}_{2}(n)\cong \operatorname{Der}_1(\mathcal{L}_{\ast}(n))\cong T_1(n).\end{align*} $$

Moreover, we have isomorphisms of graded Lie algebras

(1.1) $$ \begin{align} \mathrm{gr}(\mathcal{E}_{\ast}(n))=\bigoplus_{r\geq 1}\operatorname{gr}^r(\mathcal{E}_{\ast}(n))\cong \operatorname{Der}(\mathcal{L}_{\ast}(n))\cong \bigoplus_{r\geq 1}T_r(n) \end{align} $$

(see Section 3.5). In what follows, we identify these three graded Lie algebras.

1.2 Actions of the derivation Lie algebra on $B_d(n)$

Let $A_d(n)$ be the $\Bbbk $ -vector space spanned by Jacobi diagrams of degree d on n oriented arcs. We consider a filtration for $A_d(n)$

$$ \begin{align*}A_d(n)=A_{d,0}(n)\supset A_{d,1}(n)\supset A_{d,2}(n)\supset \cdots,\end{align*} $$

where $A_{d,k}(n)$ is the subspace of $A_d(n)$ spanned by Jacobi diagrams with at least k trivalent vertices. By restricting the functor $A_d: \mathbf {F}^{\mathrm {op}}\to \mathbf {fVect}$ that we defined in [16] to the endomorphisms, we obtain an action of $\operatorname {End}(F_n)$ on $A_d(n)$ . (See Section 2.3 and Section 4.)

Let $V_n$ be an n-dimensional $\Bbbk $ -vector space, which will be identified with the first cohomology of a handlebody of genus n. The associated graded vector space of $A_d(n)$ is isomorphic via the PBW map [3] to a graded vector space $B_d(n)=\bigoplus _{k\geq 0}B_{d,k}(n)$ of $V_n$ -colored open Jacobi diagrams of degree d, where $B_{d,k}(n)$ is the subspace of $B_d(n)$ spanned by open Jacobi diagrams with exactly k trivalent vertices.

We defined in [16] a $\mathrm {gr}(\operatorname {IA}(n))$ -action on $B_d(n)$ by using the bracket map

$$ \begin{align*}[\cdot,\cdot]:B_{d,k}(n)\otimes_{\mathbb{Z}} \operatorname{gr}^r(\operatorname{IA}(n))\rightarrow B_{d,k+r}(n). \end{align*} $$

We extend the $\mathrm {gr}(\operatorname {IA}(n))$ -action to an action of $\mathrm {gr}(\mathcal {E}_{\ast }(n))$ on $B_d(n)$ .

We define a $\Bbbk $ -linear map

$$ \begin{align*}[\cdot,\cdot]:B_{d,k}(n)\otimes_{\mathbb{Z}}\operatorname{gr}^r(\mathcal{E}_{\ast}(n))\rightarrow B_{d,k+r}(n) \end{align*} $$

by using the following theorem.

Theorem 1.1 see Theorem 4.1

For any $r\geq 1$ , we have

$$ \begin{align*}[A_{d,k}(n),\mathcal{E}_r(n)]\subset A_{d,k+r}(n).\end{align*} $$

To prove this theorem, we introduce a category $\mathbf {A}^{L}$ , which includes as full subcategories the category $\mathbf {A}$ of Jacobi diagrams in handlebodies and the category isomorphic to the PROP for Casimir Lie algebras [13]. (See Section 4 and Appendix A).

By using the bracket maps, we obtain $\Bbbk $ -linear maps

$$ \begin{align*}\tilde{\beta}_{d,k}^r: \operatorname{gr}^r(\mathcal{E}_{\ast}(n))\rightarrow\operatorname{Hom}(B_{d,k}(n),B_{d,k+r}(n)), \end{align*} $$

which form an action of the graded Lie algebra $\mathrm {gr}(\mathcal {E}_{\ast }(n))$ on the graded vector space $B_d(n)$ .

We also define a $\Bbbk $ -linear map

$$ \begin{align*}c:B_{d,k}(n)\otimes_{\mathbb{Z}} T_r(n) \rightarrow B_{d,k+r}(n), \end{align*} $$

which is an analogue of the contraction map for a vector space and its dual vector space (see Section 5). By using the map c, we obtain $\Bbbk $ -linear maps

$$ \begin{align*}\gamma_{d,k}^r: T_r(n)\rightarrow \operatorname{Hom}(B_{d,k}(n),B_{d,k+r}(n)),\end{align*} $$

which form an action of the graded Lie algebra $\bigoplus _{r\geq 1}T_r(n)$ on the graded vector space $B_d(n)$ .

Via the isomorphisms (1.1), these two actions of the derivation Lie algebra $\operatorname {Der}(\mathcal {L}_{\ast }(n))$ on $B_d(n)$ coincide up to sign. (See Theorem 6.1.)

By using the linear map c for computation, we obtain the surjectivity of the bracket map.

Proposition 1.2 see Proposition 7.8

For $n\geq 2d-k$ , the bracket map

$$ \begin{align*}[\cdot,\cdot]:B_{d,k}(n)\otimes_{\mathbb{Z}} \operatorname{gr}^1(\operatorname{IA}(n))\rightarrow B_{d,k+1}(n)\end{align*} $$

is surjective.

1.3 The $\operatorname {GL}(n;\mathbb {Z})$ -module structure of $B_d(n)$

The $\operatorname {GL}(n;\mathbb {Z})$ -action on $B_d(n)$ that is induced by the $\operatorname {Aut}(F_n)$ -action on $A_d(n)$ naturally extends to a polynomial $\operatorname {GL}(V_n)$ -action on $B_d(n)$ [16]. Therefore, the $\operatorname {GL}(V_n)$ -module $B_d(n)$ can be decomposed into the direct sum of images of the Schur functors. In general, however, it remains open to obtain an irreducible decomposition of $B_d(n)$ as $\operatorname {GL}(V_n)$ -modules. We can reduce this problem to the connected parts $B_{d,k}^c(n)\subset B_{d,k}(n)$ (see Theorem 7.2).

For a partition $\lambda \vdash N$ , let $V_{\lambda }$ denote the image of $V_n$ under the Schur functor $\mathbb {S}_{\lambda }$ . By using the results by Bar-Natan [4], we have isomorphisms of $\operatorname {GL}(V_n)$ -modules

$$ \begin{align*}B_3(n)=B_{3,0}(n)\oplus\cdots\oplus B_{3,4}(n),\end{align*} $$

where

$$ \begin{gather*} \begin{aligned} B_{3,0}(n)&\cong V_{(6)}\oplus V_{(4,2)}\oplus V_{(2^3)},\\ B_{3,1}(n)&\cong V_{(3,1^2)}\oplus V_{(2,1^3)},\\ B_{3,2}(n)&\cong V_{(4)}\oplus V_{(3,1)}\oplus (V_{(2^2)})^{\oplus 2},\\ B_{3,3}(n)&=B^c_{3,3}\cong V_{(1^3)},\\ B_{3,4}(n)&=B^c_{3,4}\cong V_{(2)} \end{aligned} \end{gather*} $$

(see Proposition 7.6 for the cases $d=3,4,5$ ).

In general degrees, we obtain irreducible decompositions of $B_{d,k}(n)$ as $\operatorname {GL}(V_n)$ -modules for $k=0,1$ .

Proposition 1.3 see Proposition 7.7

For any $d\geq 1$ , we have

$$ \begin{align*}B_{d,0}(n)\cong \bigoplus_{\lambda\vdash d} V_{2\lambda}, \end{align*} $$

where $2\lambda =(2\lambda _1,\cdots ,2\lambda _r)\vdash 2d$ for $\lambda =(\lambda _1,\cdots ,\lambda _r)\vdash d$ . For any $d\geq 2$ , we have

$$ \begin{align*}B_{d,1}(n)\cong \bigoplus_{\lambda\vdash 2d-1 \text{ with exactly }3\text{ odd parts}}V_{\lambda}. \end{align*} $$

1.4 The $\operatorname {Aut}(F_n)$ -module structure of $A_d(n)$

We consider the $\operatorname {Aut}(F_n)$ -module structure of $A_d(n)$ and give an indecomposable decomposition of $A_d(n)$ . We have

$$ \begin{align*}A_0(n)=\Bbbk \quad(n\geq 0),\quad A_d(0)=0 \quad (d\geq 1)\end{align*} $$

and we studied the cases where $d=1, 2$ in [16]. Thus, we mainly consider the cases where $d\geq 3,n\geq 1$ .

For $X\in A_d(2d)$ , let

$$ \begin{align*}A_d X:\mathbf{F}^{\mathrm{op}}\rightarrow \mathbf{fVect}\end{align*} $$

denote the subfunctor of $A_d$ generated by X. That is, for any $n\in \mathbb {N}$ , $A_d X(n)$ is the $\operatorname {Aut}(F_n)$ -submodule of $A_d(n)$ defined by

$$ \begin{align*}A_d X (n):=\operatorname{Span}_\Bbbk\{A_d(f)(X)\mid f\in\mathbf{F}^{\mathrm{op}}(2d,n)\}.\end{align*} $$

Set

Then, we have the following direct decomposition of $A_d(n)$ as $\operatorname {Aut}(F_n)$ -modules, which is indecomposable for $n\geq 2d$ .

Theorem 1.4 see Theorems 8.2, 8.9

We have $A_d(n)=A_d P(n)\oplus A_d Q(n)$ for any $d,n\geq 1$ . This direct decomposition is indecomposable for $n\geq 2d$ .

In degree $1$ , we have $A_1 Q(n)=0$ and $A_1(n)\cong \operatorname {Sym}^2(V_n)$ is simple for $n\geq 1$ . In [16], we obtained that the direct decomposition of $A_2(n)$ is indecomposable for $n\geq 3$ (see Theorem 6.9 of [16]). We improve Theorem 1.4 for $d=3,4$ (see Theorems 8.12 and 8.17).

In general degree d, we obtain the radical of $A_{d,k}(n)$ for any $k\geq 0$ if $n\geq 2d$ .

Theorem 1.5 see Theorem 8.6

Let $n\geq 2d$ . The filtration of $A_d(n)$ by the number of trivalent vertices coincides with the radical filtration of $A_d(n)$ .

In degree $3$ , we obtain the socle of $A_3(n)$ as well (see Proposition 8.15).

1.5 Direct decomposition of the functor $A_d$

Lastly, we give an indecomposable decomposition of the functor $A_d$ .

By Theorem 1.4, we obtain an indecomposable decomposition of the functor $A_d$ .

Theorem 1.6 see Theorem 10.1

We have an indecomposable decomposition

(1.2) $$ \begin{align} A_d=A_d P\oplus A_d Q \end{align} $$

in the functor category $\mathbf {fVect}^{\mathbf {F}^{\mathrm {op}}}$ .

In degree $1$ , we have $A_1 Q=0$ and $A_1=A_1 P$ . In [16], we obtained the direct decomposition (1.2) of the functor $A_2$ and proved that equation (1.2) is indecomposable (see Proposition 6.5 and Theorem 6.14 of [16]).

1.6 Organization of the paper

In Section 2, we recall the category $\mathbf {A}$ of Jacobi diagrams in handlebodies, N-series and graded Lie algebras, contents of the previous paper [16], Hopf algebras and Lie algebras in a linear symmetric strict monoidal category. In Section 3, we construct the Andreadakis filtration and the Johnson homomorphism of $\operatorname {End}(F_n)$ . In Section 4, we construct an action of the derivation Lie algebra $\operatorname {Der}(\mathcal {L}_{\ast }(n))$ on $B_d(n)$ , which is defined by the bracket map. In preparation for the definition of the bracket map, we construct an extended category $\mathbf {A}^{L}$ of the category $\mathbf {A}$ , which includes a Lie algebra structure. In Section 5, we define a contraction map, which forms another action of $\operatorname {Der}(\mathcal {L}_{\ast }(n))$ on $B_d(n)$ . In Section 6, we prove that two actions of $\operatorname {Der}(\mathcal {L}_{\ast }(n))$ on $B_d(n)$ defined in Sections 4 and 5 coincide up to sign. In Section 7, we compute the $\operatorname {GL}(n;\mathbb {Z})$ -module structure of $B_d(n)$ . In Section 8, we study the $\operatorname {Aut}(F_n)$ -module structure of $A_d(n)$ by using the $\operatorname {GL}(n;\mathbb {Z})$ -module structure of $B_d(n)$ and the action of $\operatorname {Der}(\mathcal {L}_{\ast }(n))$ on $B_d(n)$ . In Section 10, we give an indecomposable decomposition of the functor $A_d$ . In Appendix A, we study an expected presentation of the category $\mathbf {A}^{L}$ .

2 Preliminaries

In this section, we recall the contents of the previous paper [16] and definitions of the category $\mathbf {A}$ of Jacobi diagrams in handlebodies, Hopf algebras and Lie algebras in a symmetric strict monoidal category and an action of an N-series on a filtered vector space and that of a graded Lie algebra on a graded vector space.

In what follows, we work over a fixed field $\Bbbk $ of characteristic $0$ . For a vector space V and an abelian group G, we just write $V\otimes G$ instead of $V\otimes _{\mathbb {Z}} G$ . For vector spaces V and W, we also write $V\otimes W$ instead of $V\otimes _{\Bbbk } W$ .

For $n\geq 0$ , let $[n]:=\{1,\cdots ,n\}$ .

2.1 The category $\mathbf {A}$ of Jacobi diagrams in handlebodies

Here, we briefly review the category $\mathbf {A}$ of Jacobi diagrams in handlebodies defined in [11]. We use the same notations as in [16].

For $n\geq 0$ , let be the oriented $1$ -manifold consisting of n arc components.

Let $I=[-1,1]$ . For $n\geq 0$ , let $U_n \subset \mathbb {R}^3$ denote the handlebody of genus n that is obtained from the cube $I^3$ by attaching n handles on the top square $I^2 \times \{1\}$ as depicted in Figure 1. We call $l := I \times \{0\} \times \{-1\}$ the bottom line of $U_n$ and $l':= I\times \{0\} \times \{1\}$ the upper line of $U_n$ . We call $S := I^2 \times \{-1\}$ the bottom square of $U_n$ .

Figure 1 The handlebody $U_n$ .

For $i\in [n]$ , let $x_i$ be a loop which goes through only the i-th handle of the handlebody $U_n$ just once, and let $x_i$ denote its homotopy class as well. In what follows, for loops $\gamma _1$ and $\gamma _2$ with base points on l, let $\gamma _2\gamma _1$ denote the loop that goes through $\gamma _1$ first and then goes through $\gamma _2$ . That is, we write a product of elements of the fundamental group of $U_n$ in the opposite order to the usual one. Let $H=H_1(U_n;\mathbb {Z})$ , and let $\bar {x}_i\in H$ be the homology class of $x_i$ . We have $H=\bigoplus _{i=1}^n \mathbb {Z}\bar {x}_i$ and $\pi _1(U_n)=\langle x_1,\cdots ,x_n\rangle $ . Let

$$ \begin{align*}V_n=H^1(U_n;\Bbbk)=\operatorname{Hom}(H,\Bbbk),\end{align*} $$

and let $\{v_1,\cdots ,v_n\}$ be the dual basis of $\{\bar {x}_1,\cdots ,\bar {x}_n\}$ .

The objects in $\mathbf {A}$ are nonnegative integers.

For $m,n\geq 0$ , the hom-set $\mathbf {A}(m,n)$ is the $\Bbbk $ -vector space spanned by $(m,n)$ -Jacobi diagrams modulo the STU relation. An $(m,n)$ -Jacobi diagram is a Jacobi diagram on $X_n$ mapped into $U_m$ in such a way that the endpoints of $X_n$ are uniformly distributed on the bottom line l of $U_m$ (see [11, 16] for further details). We usually depict $(m,n)$ -Jacobi diagrams by drawing their images under the orthogonal projection of $\mathbb {R}^3$ onto $\mathbb {R}\times \{0\}\times \mathbb {R}$ .

The degree of an $(m,n)$ -Jacobi diagram is the degree of its Jacobi diagram. Let $\mathbf {A}_d(m,n)\subset \mathbf {A}(m,n)$ be the subspace spanned by $(m,n)$ -Jacobi diagrams of degree d. We have $\mathbf {A}(m,n)=\bigoplus _{d\geq 0} \mathbf {A}_d(m,n)$ .

The category $\mathbf {A}$ has a structure of a linear symmetric strict monoidal category. The tensor product on objects is addition. The monoidal unit is $0$ . The tensor product on morphisms is juxtaposition followed by horizontal rescaling and relabelling of indices. The symmetry is determined by

2.2 N-series and graded Lie algebras

Here, we briefly review the definition of an action of an N-series on a filtered vector space and the induced action of the graded Lie algebra on the graded vector space (see [16] for details).

An N-series $K_{\ast }=(K_n)_{n\geq 1}$ of a group K is a descending series

$$ \begin{align*}K=K_1\supset K_2\supset \cdots \end{align*} $$

such that $[K_n,K_m]\subset K_{n+m}$ for all $n,m\geq 1$ .

A morphism $f: G_{\ast }\rightarrow K_{\ast }$ between N-series is a group homomorphism $f: G_1 \rightarrow K_1$ such that we have $f(G_n)\subset K_n$ for all $n\geq 1.$

For a filtered vector space $W_{\ast }$ , set

$$ \begin{align*}\operatorname{Aut}_n(W_{\ast}):=\{\phi\in\operatorname{Aut}_{\mathbf{fVect}}(W_{\ast})\mid [\phi,w]\in W_{k+n}\text{ for all }w\in W_k, k\geq 0\}\quad (n\geq 1), \end{align*} $$

where $[\phi ,w]:=\phi (w)-w$ for $w\in W_k$ . We can easily check that $\operatorname {Aut}_{\ast }(W_{\ast }):=(\operatorname {Aut}_n(W_{\ast }))_{n\geq 1}$ is an N-series.

Definition 2.1. (Action of N-series on filtered vector spaces) Let $K_{\ast }$ be an N-series and $W_{\ast }$ be a filtered vector space. An action of $K_{\ast }$ on $W_{\ast }$ is a morphism $f:K_{\ast }\rightarrow \operatorname {Aut}_{\ast }(W_{\ast })$ between N-series.

For an N-series $K_{\ast }$ , we have a graded Lie algebra $\mathrm {gr}(K_{\ast })=\bigoplus _{n\geq 1} K_n/K_{n+1}$ , where the Lie bracket is defined by the commutator.

For a graded vector space $W=\bigoplus _{k\geq 0}W_k$ , set

$$ \begin{align*}\operatorname{End}_n(W):=\{\phi\in\operatorname{End}(W)\mid \phi(W_k)\subset W_{k+n}\text{ for }k\geq 0\}\quad(n\geq 1). \end{align*} $$

We can check that $\operatorname {End}_{+}(W)=\bigoplus _{n\geq 1}\operatorname {End}_n(W)$ is a graded Lie algebra, where the Lie bracket is defined by

$$ \begin{align*}[f,g]:=f\circ g-g\circ f \quad \text{for} \quad f\in\operatorname{End}_k(W),g\in\operatorname{End}_l(W)\; (k,l\geq 1).\end{align*} $$

Definition 2.2. (Action of graded Lie algebras on graded vector spaces) Let $L_{+}=\bigoplus _{n\geq 1}L_n$ be a graded Lie algebra and $W=\bigoplus _{k\geq 0}W_k$ be a graded vector space. An action of $L_{+}$ on W is a morphism $f:L_{+}\rightarrow \operatorname {End}_{+}(W)$ between graded Lie algebras.

Proposition 2.3. An action of an N-series $K_{\ast }$ on a filtered vector space $W_{\ast }$ induces an action of the graded Lie algebra $\mathrm {gr}(K_{\ast })$ on the graded vector space $\mathrm {gr}(W_{\ast })$ , which is a morphism

$$ \begin{align*}\rho_{+}:\bigoplus_{n\geq 1}\operatorname{gr}^n(K_{\ast}) \rightarrow\bigoplus_{n\geq 1}\operatorname{End}_n(\mathrm{gr}(W_{\ast}))\end{align*} $$

defined by $\rho _{+}(gK_{n+1})([v]_{W_{k+1}})=[[g,v]]_{W_{k+n+1}}$ for $gK_{n+1}\in \operatorname {gr}^n(K_{\ast })$ , $[v]_{W_{k+1}}\in \operatorname {gr}^k(W_{\ast })$ .

The proof can be seen in Proposition 5.14 of [16].

2.3 Contents of the previous paper

Here, we briefly review the notations and contents of the previous paper [16]. Let $\operatorname {Aut}(F_n)$ denote the automorphism group of the free group $F_n$ of rank n and $\operatorname {GL}(n;\mathbb {Z})$ the general linear group of degree n. Let $\operatorname {IA}(n)$ denote the IA-automorphism group of $F_n$ , that is the kernel of the canonical surjection

$$ \begin{align*}\operatorname{Aut}(F_n)\rightarrow \operatorname{Aut}(H_1(F_n;\mathbb{Z}))\cong \operatorname{GL}(n;\mathbb{Z}).\end{align*} $$

Let $\Gamma _{\ast }(\operatorname {IA}(n))=(\Gamma _{r}(\operatorname {IA}(n)))_{r\geq 1}$ denote the lower central series of $\operatorname {IA}(n)$ , and $\mathrm {gr}(\operatorname {IA}(n))=\bigoplus _{r\geq 1}\operatorname {gr}^r(\operatorname {IA}(n))$ the associated graded Lie algebra, where $\operatorname {gr}^r(\operatorname {IA}(n))=\Gamma _{r}(\operatorname {IA}(n))/\Gamma _{r+1}(\operatorname {IA}(n))$ .

Let $A_d(n)=\mathbf {A}_d(0,n)$ denote the $\Bbbk $ -vector space of Jacobi diagrams of degree d on $X_n$ . We consider a filtration for $A_d(n)$

$$ \begin{align*}A_d(n)=A_{d,0}(n)\supset A_{d,1}(n)\supset\cdots\supset A_{d,2d-2}(n)\supset A_{d,2d-1}(n)=0 \end{align*} $$

such that $A_{d,k}(n)\subset A_d(n)$ is the subspace spanned by Jacobi diagrams with at least k trivalent vertices. Hence, $A_d(n)$ is a filtered vector space.

Let $\mathbf {F}$ denote the category of finitely generated free groups and $\mathbf {fVect}$ the category of filtered vector spaces over $\Bbbk $ .

We have a $\Bbbk $ -vector space isomorphism

$$ \begin{align*}Z:\Bbbk\mathbf{F}^{\mathrm{op}}(m,n)\xrightarrow{\cong} \mathbf{A}_0(m,n)\end{align*} $$

from the hom-set $\Bbbk \mathbf {F}^{\mathrm {op}}(m,n)$ of the $\Bbbk $ -linearization of the opposite category of $\mathbf {F}$ to the degree $0$ part of the hom-set $\mathbf {A}(m,n)$ [11]. We define a functor

$$ \begin{align*}A_d:\mathbf{F}^{\mathrm{op}}\rightarrow \mathbf{fVect}\end{align*} $$

by $A_d(n)=\mathbf {A}_d(0,n)$ for an object $n\in \mathbb {N}$ and $A_d(f)=Z(f)_{\ast }$ for a morphism $f\in \mathbf {F}^{\mathrm {op}}(m,n)$ , where $Z(f)_{\ast }$ denotes the post-composition with $Z(f)$ . The functor $A_d$ is a polynomial functor of degree $2d$ in the sense of [12, 20] (see Remark 3.1 of [16]). By restricting this functor to the automorphism group, we obtain an action of the opposite group $\operatorname {Aut}(F_n)^{\mathrm {op}}$ of $\operatorname {Aut}(F_n)$ on $A_d(n)$ for each $n\geq 0$ . We consider this action as a right action of $\operatorname {Aut}(F_n)$ on $A_d(n)$ . The $\operatorname {Aut}(F_n)$ -action on $A_d(n)$ induces an action on $A_d(n)$ of the outer automorphism group $\operatorname {Out}(F_n)$ of $F_n$ (see Theorem 5.1 in [16]).

On the other hand, the associated graded vector space $\mathrm {gr}(A_d(n))$ of $A_d(n)$ is identified via the PBW map [2, 3]

(2.1) $$ \begin{align} \theta_{d,n}:\mathrm{gr}(A_d(n))\xrightarrow{\cong} B_d(n) \end{align} $$

with the graded $\Bbbk $ -vector space $B_d(n)=\bigoplus _{k\geq 0}B_{d,k}(n)= \bigoplus _{k=0}^{2d-2}B_{d,k}(n)$ of $V_n$ -colored open Jacobi diagrams of degree d, where the grading is determined by the number of trivalent vertices. Note that we have $\theta _{d,n}=\bigoplus _{k}\theta _{d,n,k}$ , where

$$ \begin{align*}\theta_{d,n,k}:\operatorname{gr}^k(A_d(n))\xrightarrow{\cong} B_{d,k}(n).\end{align*} $$

Let $\mathbf {FAb}$ denote the category of finitely generated free abelian groups and $\mathbf {gVect}$ the category of graded vector spaces over $\Bbbk $ .

We define a functor

$$ \begin{align*}B_d: \mathbf{FAb}^{\mathrm{op}}\rightarrow \mathbf{gVect}\end{align*} $$

by sending an object $n\in \mathbb {N}$ to the graded vector space $B_d(n)$ and a morphism $f\in \mathbf {FAb}^{\mathrm {op}}(m,n)=\operatorname {Mat}(m,n;\mathbb {Z})$ to $B_d(f)$ , which is a right action on each coloring, where we consider an element of $V_n$ as a $(1\times n)$ -matrix. By restricting this functor to the automorphism group, we obtain an action of the opposite group $\operatorname {GL}(n;\mathbb {Z})^{\mathrm {op}}$ of $\operatorname {GL}(n;\mathbb {Z})$ on $B_d(n)$ for each $n\geq 0$ . We consider this action as a right action of $\operatorname {GL}(n;\mathbb {Z})$ on $B_d(n)$ . Note that the $\operatorname {GL}(n;\mathbb {Z})$ -action on $B_d(n)$ naturally extends to a $\operatorname {GL}(V_n)$ -action on $B_d(n)$ .

Proposition 2.4 see Proposition 3.2 of [16]

For $d\geq 0$ , the PBW maps equation (2.1) give a natural isomorphism

$$ \begin{align*}\theta_d:\mathrm{gr}\circ A_d\overset{\cong}\Rightarrow B_d\circ\mathrm{ab}^{\mathrm{op}},\end{align*} $$

where $\mathrm {ab}^{\mathrm {op}}$ denotes the opposite functor of the abelianization functor and $\mathrm {gr}$ denote the functor that sends a filtered vector space to its associated graded vector space.

By this proposition, it turns out that the $\operatorname {Aut}(F_n)$ -action on $A_d(n)$ , which is an action of an extended N-series on a filtered vector space, induces two actions on $B_d(n)$ , which form an action of an extended graded Lie algebra on a graded vector space (see Theorem 5.15 of [16] and [10] for extended N-series and extended graded Lie algebras). One of them is the $\operatorname {GL}(n;\mathbb {Z})$ -action, and the other of them is an action of the graded Lie algebra $\mathrm {gr}(\operatorname {IA}(n))$ on the graded vector space $B_d(n)$ , which consists of $\operatorname {GL}(n;\mathbb {Z})$ -module homomorphisms

(2.2) $$ \begin{align} [\cdot,\cdot]: B_{d,k}(n)\otimes \operatorname{gr}^r(\operatorname{IA}(n)) \rightarrow B_{d,k+r}(n) \end{align} $$

for $k\geq 0, r\geq 1$ (see Proposition 5.10 and Theorem 5.15 of [16]). By using these two actions on $B_d(n)$ , we obtained an indecomposable decomposition of $A_2(n)$ as $\operatorname {Aut}(F_n)$ -modules (see Theorem 6.9 of [16]).

2.4 Hopf algebra in a symmetric strict monoidal category

We review the definition of a Hopf algebra in a symmetric strict monoidal category. Let $\mathcal {C}=(\mathcal {C},\otimes ,I,P)$ be a symmetric strict monoidal category. A Hopf algebra in $\mathcal {C}$ is an object H in $\mathcal {C}$ equipped with morphisms

$$ \begin{align*}\mu:H\otimes H\rightarrow H, \quad \eta:I\rightarrow H, \quad \Delta:H\rightarrow H\otimes H, \quad \epsilon:H\rightarrow I,\quad S:H\rightarrow H, \end{align*} $$

called the multiplication, unit, comultiplication, counit and antipode, respectively, satisfying

1. (1) $\mu (\mu \otimes \operatorname {id}_H)=\mu (\operatorname {id}_H \otimes \mu ),\quad \mu (\eta \otimes \operatorname {id}_H)=\operatorname {id}_H=\mu (\operatorname {id}_H\otimes \eta ),$

2. (2) $(\Delta \otimes \operatorname {id}_H)\Delta =(\operatorname {id}_H\otimes \Delta )\Delta ,\quad (\epsilon \otimes \operatorname {id}_H)\Delta =\operatorname {id}_H=(\operatorname {id}_H\otimes \epsilon )\Delta ,$

3. (3) $\epsilon \eta =\operatorname {id}_I, \quad \epsilon \mu =\epsilon \otimes \epsilon ,\quad \Delta \eta =\eta \otimes \eta ,$

4. (4) $\Delta \mu =(\mu \otimes \mu )(\operatorname {id}_H\otimes P_{H,H}\otimes \operatorname {id}_H)(\Delta \otimes \Delta ),$

5. (5) $\mu (\operatorname {id}_H\otimes S)\Delta =\mu (S\otimes \operatorname {id}_H)\Delta =\eta \epsilon .$

A Hopf algebra H is said to be cocommutative if $P_{H,H}\Delta =\Delta .$

Define $\mu _n : H^{\otimes {n}} \otimes H^{\otimes {n}} \rightarrow H^{\otimes {n}}$ and $\Delta _m : H^{\otimes {m}} \rightarrow H^{\otimes {m}} \otimes H^{\otimes {m}}$ inductively by

$$ \begin{align*}\mu_0=\operatorname{id}_I, \quad \mu_{n+1}=(\mu_n \otimes \mu)(\operatorname{id}_{H^{\otimes{n}}} \otimes P_{H,H^{\otimes{n}}} \otimes \operatorname{id}_H)\end{align*} $$

for $n\geq 0$ and by

$$ \begin{align*}\Delta_0=\operatorname{id}_I,\quad \Delta_{m+1}=(\operatorname{id}_{H^{\otimes{m}}} \otimes P_{H^{\otimes{m}},H} \otimes \operatorname{id}_H)(\Delta_m \otimes \Delta) \end{align*} $$

for $m\geq 0$ .

For morphisms $f,\ f': H^{\otimes {m}} \rightarrow H^{\otimes {n}}$ , $m,n\geq 0$ , the convolution $f\ast f'$ of f and $f'$ is defined by

$$ \begin{align*}f\ast f':= \mu_n (f\otimes f')\Delta_m. \end{align*} $$

The category $\mathbf {A}$ has a cocommutative Hopf algebra with the object $1$ , where

2.5 Lie algebra in a linear symmetric strict monoidal category

We review the definition of a Lie algebra in a linear symmetric strict monoidal category. Let $\mathcal {C}=(\mathcal {C},\otimes ,I,P)$ be a linear symmetric strict monoidal category. A Lie algebra in $\mathcal {C}$ is an object L in $\mathcal {C}$ equipped with a morphism

$$ \begin{align*}[\cdot,\cdot]:L\otimes L\rightarrow L\end{align*} $$

satisfying

1. (1) $[\cdot ,\cdot ](\operatorname {id}_{L\otimes L}+P_{L,L})=0$ ,

2. (2) $[\cdot ,\cdot ](\operatorname {id}_L\otimes [\cdot ,\cdot ])(\operatorname {id}_{L^{\otimes 3}}+\sigma +\sigma ^2)=0,$ where $\sigma =(1,2,3):L^{\otimes 3}\rightarrow L^{\otimes 3}$ .

3 Andreadakis filtration $\mathcal {E}_{\ast }(n)$ of $\operatorname {End}(F_n)$

We briefly review the Andreadakis filtration and the Johnson homomorphism of $\operatorname {Aut}(F_n)$ . See [22] for further details. Then we consider its extension to the endomorphism monoid $\operatorname {End}(F_n)$ of $F_n$ .

3.1 Andreadakis filtration $\mathcal {A}_{\ast }(n)$ of $\operatorname {Aut}(F_n)$

In what follows, we consider the left action of $\operatorname {Aut}(F_n)$ on $F_n$ . Let $\Gamma _r:=\Gamma _r(F_n)$ denote the r-th term of the lower central series of the free group $F_n$ of rank n. Let $\mathcal {L}_r(n) :=\Gamma _r/\Gamma _{r+1}$ for $r\geq 1$ . Note that $H=\mathcal {L}_1(n)$ and that $\mathcal {L}_r(n)$ is the degree r part of the free Lie algebra $\mathcal {L}_{\ast }(n)$ on H.

For $r\geq 0$ , the left action of $\operatorname {Aut}(F_n)$ on each nilpotent quotient $F_n/\Gamma _{r+1}$ induces a group homomorphism

$$ \begin{align*}\operatorname{Aut}(F_n)\rightarrow \operatorname{Aut}(F_n/\Gamma_{r+1}).\end{align*} $$

Set

$$ \begin{align*}\mathcal{A}_{r}(n) := \ker(\operatorname{Aut}(F_n)\rightarrow \operatorname{Aut}(F_n/\Gamma_{r+1})) \lhd \operatorname{Aut}(F_n).\end{align*} $$

Then we have a filtration, which is called the Andreadakis filtration of $\operatorname {Aut}(F_n)$ :

$$ \begin{align*}\operatorname{Aut}(F_n)=\mathcal{A}_0(n)\supset\mathcal{A}_1(n)=\operatorname{IA}(n)\supset\mathcal{A}_2(n)\supset\cdots.\end{align*} $$

For $r\geq 1$ , the Johnson homomorphism

$$ \begin{align*}\tau_r:\mathrm{gr}^r(\mathcal{A}_{\ast}(n))\hookrightarrow \operatorname{Hom}(H,\mathcal{L}_{r+1}(n))\end{align*} $$

is the injective homomorphism induced by the group homomorphism

$$ \begin{align*}\tau^{\prime}_{r}:\mathcal{A}_r(n)\rightarrow\operatorname{Hom}(H,\mathcal{L}_{r+1}(n))\end{align*} $$

defined by

$$ \begin{align*}\tau^{\prime}_{r}(f)(x\Gamma_2):=f(x)x^{-1}\Gamma_{r+2}\quad\text{for}\ f\in\mathcal{A}_r(n), x\in F_n.\end{align*} $$

3.2 The target group of the Johnson homomorphism

The target group $\operatorname {Hom}(H,\mathcal {L}_{r+1}(n))\cong H^{\ast }\otimes \mathcal {L}_{r+1}(n)$ of the Johnson homomorphism is identified with the degree r part $\operatorname {Der}_r(\mathcal {L}_{\ast }(n))$ of the derivation Lie algebra $\operatorname {Der}(\mathcal {L}_{\ast }(n))$ of the free Lie algebra $\mathcal {L}_{\ast }(n)$ and with the tree module $T_r(n)$ via abelian group isomorphisms

(3.1) $$ \begin{align} H^{\ast}\otimes \mathcal{L}_{r+1}(n)\cong \operatorname{Der}_r(\mathcal{L}_{\ast}(n)) \cong T_r(n). \end{align} $$

Here, we briefly review the derivation Lie algebra and the tree module. (See [22] for details.)

A derivation f of $\mathcal {L}_{\ast }(n)$ is a $\mathbb {Z}$ -linear map $f:\mathcal {L}_{\ast }(n)\rightarrow \mathcal {L}_{\ast }(n)$ such that $f([a,b])=[f(a),b]+[a,f(b)]$ for any $a,b\in \mathcal {L}_{\ast }(n)$ . The derivation Lie algebra $\operatorname {Der}(\mathcal {L}_{\ast }(n))$ of the Lie algebra $\mathcal {L}_{\ast }(n)$ is the set of all derivations of $\mathcal {L}_{\ast }(n)$ . The degree r part $\operatorname {Der}_r(\mathcal {L}_{\ast }(n))$ of the derivation Lie algebra is defined to be

$$ \begin{align*}\operatorname{Der}_r(\mathcal{L}_{\ast}(n))=\{f\in \operatorname{Der}(\mathcal{L}_{\ast}(n))\mid f(a)\in \mathcal{L}_{r+1}(n) \text{ for any }a\in H\}.\end{align*} $$

Then we have $\operatorname {Der}(\mathcal {L}_{\ast }(n))=\bigoplus _{r\geq 0}\operatorname {Der}_r(\mathcal {L}_{\ast }(n))$ and abelian group isomorphisms

$$ \begin{align*}\operatorname{Der}_r(\mathcal{L}_{\ast}(n))\cong \operatorname{Hom}(H,\mathcal{L}_{r+1}(n))\cong H^{\ast}\otimes \mathcal{L}_{r+1}(n).\end{align*} $$

We call a connected Jacobi diagram with no cycle a trivalent tree. For $r\geq 0$ , a trivalent tree is called a rooted trivalent tree of degree r if it has one univalent vertex (called the root) that is colored by an element of $H^{\ast }$ and $r+1$ univalent vertices (called leaves) that are colored by elements of H. Let $T_r(n)$ denote the $\mathbb {Z}$ -module spanned by rooted trivalent trees of degree r modulo the AS, IHX and multilinearity relations. We have an abelian group isomorphism

$$ \begin{align*}\Phi:H^{\ast}\otimes\mathcal{L}_{r+1}(n)\xrightarrow{\cong} T_{r}(n)\end{align*} $$

defined by

for $v_i\in H^{\ast }$ , $[\bar {x}_{i_1},\cdots ,[\bar {x}_{i_r},\bar {x}_{i_{r+1}}]\cdots ]\in \mathcal {L}_{r+1}(n)$ .

3.3 Andreadakis filtration $\mathcal {E}_{\ast }(n)$ of $\operatorname {End}(F_n)$

We extend the above construction to the endomorphism monoid $\operatorname {End}(F_n)$ of $F_n$ . For $r\geq 0$ , consider the canonical map

$$ \begin{align*}\rho_r:\operatorname{End}(F_n)\rightarrow \operatorname{End}(F_n/\Gamma_{r+1})\end{align*} $$

and set $\mathcal {E}_r(n):=\ker (\rho _r)$ . Then we have a filtration of monoids

$$ \begin{align*}\operatorname{End}(F_n)=\mathcal{E}_0(n)\supset\mathcal{E}_1(n)\supset\cdots,\end{align*} $$

and we call $\mathcal {E}_{\ast }(n)=(\mathcal {E}_{r}(n))_{r\geq 0}$ the Andreadakis filtration of $\operatorname {End}(F_n)$ .

For $f\in \operatorname {End}(F_n)$ and $x,y\in F_n$ , set

$$ \begin{align*}[f,x]:=f(x)x^{-1},\quad {}^y x=yxy^{-1},\end{align*} $$

and for a subset $T\subset F_n$ , set

$$ \begin{align*}[f,T]=\{[f,x]\in F_n \mid x\in T\}.\end{align*} $$

We can easily check the following lemma.

Lemma 3.1.

$$ \begin{align*}f\in \mathcal{E}_{r}(n)\quad \Leftrightarrow\quad [f, F_n]\subset \Gamma_{r+1}\quad \Leftrightarrow\quad [f,x_i]\in \Gamma_{r+1}\; (\text{for any }i\in [n]).\end{align*} $$

For subsets $S\subset \operatorname {End}(F_n)$ and $T\subset F_n$ , let $[S,T]$ denote the subgroup of $F_n$ generated by the elements $[f,x]$ for $f\in S, x\in T$ .

Lemma 3.2. We have

$$ \begin{align*}[\mathcal{E}_r(n),\Gamma_k]\subset \Gamma_{k+r} \end{align*} $$

for $r\geq 0, k\geq 1$ .

Proof. It is well known that $[\mathcal {A}_r(n),\Gamma _k]\subset \Gamma _{k+r}$ by Andreadakis [1]. The same proof can be applied to $\mathcal {E}_r(n)$ . We use induction on k. When $k=1$ , we have $[\mathcal {E}_r(n),F_n]\subset \Gamma _{r+1}$ by the definition of $\mathcal {E}_r(n)$ . Suppose that $[\mathcal {E}_r(n),\Gamma _{k-1}]\subset \Gamma _{k-1+r}$ . We will show that $[\mathcal {E}_r(n),\Gamma _{k}]\subset \Gamma _{k+r}$ . Let $f\in \mathcal {E}_r(n)$ . Recall that $\Gamma _k$ is generated by the commutator $[x,y]$ with $x\in \Gamma _{k-1},y\in F_n$ . We can check that for $x\in \Gamma _{k-1},y\in F_n$ , we have

$$ \begin{align*}[f,[x,y]]={}^{[f,y]}([[f,y]^{-1},f(x)]\cdot [[f,x],{}^{x}y]\cdot[[x,y],[f,y]^{-1}])\in \Gamma_{k+r}. \end{align*} $$

For $z,w\in \Gamma _k$ , we have

$$ \begin{gather*} [f,zw]=[f,z]\cdot{}^{z}[f,w]\equiv [f,z][f,w] \quad(\bmod\:\Gamma_{k+r+1}), \end{gather*} $$

and by letting $w=z^{-1}$ , we have

$$ \begin{gather*} [f,z^{-1}]\equiv [f,z]^{-1}\quad(\bmod\:\Gamma_{k+r+1}). \end{gather*} $$

Therefore, we have $[f,z]\in \Gamma _{k+r}$ for any $z\in \Gamma _k$ .

Define a map

$$ \begin{align*}\sigma:\operatorname{End}(F_n)\rightarrow \operatorname{End}(F_n)\end{align*} $$

by $\sigma (f)=\tilde {f}$ for $f\in \operatorname {End}(F_n)$ , where

$$ \begin{align*}\tilde{f}(x_i)=[f,x_i]^{-1}x_i=x_i f(x_i)^{-1} x_i\end{align*} $$

for $i\in [n]$ .

Lemma 3.3. We have

(3.2) $$ \begin{align} \sigma^2=\operatorname{id}_{\operatorname{End}(F_n)} \end{align} $$

(3.3) $$ \begin{align} f\in \mathcal{E}_r(n)\quad\Rightarrow\quad \sigma(f)\in \mathcal{E}_r(n) \end{align} $$

(3.4) $$ \begin{align} f\in \mathcal{E}_r(n)\quad\Rightarrow\quad f\sigma(f),\: \sigma(f) f\in \mathcal{E}_{2r}(n). \end{align} $$

Proof. We have equation (3.2) since for any $f\in \operatorname {End}(F_n)$ and $i\in [n]$ , we have

$$ \begin{align*}\sigma^2(f)(x_i)=x_i\tilde{f}(x_i)^{-1}x_i=x_ix_i^{-1}f(x_i)x_i^{-1}x_i=f(x_i).\end{align*} $$

We have equation (3.3) since, for any $f\in \mathcal {E}_r(n)$ and $i\in [n]$ , we have

$$ \begin{align*}[\tilde{f}, x_i]=[f,x_i]^{-1}\in \Gamma_{r+1}. \end{align*} $$

We prove equation (3.4). Let $f\in \mathcal {E}_r(n)$ . We have

$$ \begin{align*}[f\tilde{f},x_i]=f([\tilde{f},x_i])[f,x_i]=f([f,x_i]^{-1})[f,x_i]=[f,[f,x_i]^{-1}]\in \Gamma_{2r+1}\end{align*} $$

for any $i\in [n]$ . Thus, we have

(3.5) $$ \begin{align} f\tilde{f}\in\mathcal{E}_{2r}(n). \end{align} $$

By equation (3.3), we have $\tilde {f}\in \mathcal {E}_r(n)$ , and by equations (3.2) and (3.5),

$$ \begin{align*}\tilde{f}f=\tilde{f}\tilde{\tilde{f}}\in\mathcal{E}_{2r}(n).\end{align*} $$

For $N\geq r\geq 0$ , we define an equivalence relation $\sim _{N}$ on the monoid $\mathcal {E}_r(n)$ by

$$ \begin{align*}f\sim_{N}g \quad \overset{\text{def}}\Leftrightarrow \quad [f,x]\equiv[g,x]\; (\bmod\: \Gamma_{N+1}) \quad \text{ for any }x\in F_n \end{align*} $$

for $f,g\in \mathcal {E}_r(n)$ . Thus, we have

$$ \begin{align*}f\sim_{N} \operatorname{id}_{F_n} \quad \Leftrightarrow \quad [f,x]\in \Gamma_{N+1} \quad \text{ for any }x\in F_n \quad \Leftrightarrow \quad f\in \mathcal{E}_N(n).\end{align*} $$

Lemma 3.4. Let $r\geq 1$ . For $f\in \mathcal {E}_r(n)$ , define $f^{R}_{N}$ and $f^{L}_{N}$ for $N\geq r+1$ inductively by

$$ \begin{align*}f^{R}_{N}= \begin{cases} \tilde{f} & (N=r+1)\\ f^{R}_{N-1}\widetilde{f f^{R}_{N-1}} & (N\geq r+2), \end{cases} \end{align*} $$

$$ \begin{align*}f^{L}_{N}= \begin{cases} \tilde{f} & (N=r+1)\\ \widetilde{f^{L}_{N-1} f}f^{L}_{N-1} & (N\geq r+2). \end{cases} \end{align*} $$

Then we have

$$ \begin{gather*} f^{R}_{N}\in \mathcal{E}_r(n),\quad f f^{R}_{N}\in \mathcal{E}_N(n),\quad f^{R}_{N}\sim_{N-1} f^{R}_{N-1},\\ f^{L}_{N}\in \mathcal{E}_r(n),\quad f^{L}_{N} f\in \mathcal{E}_N(n),\quad f^{L}_{N}\sim_{N-1} f^{L}_{N-1}. \end{gather*} $$

Proof. We use induction on $N\geq r+1$ . When $N=r+1$ , by Lemma 3.3, we have $\tilde {f}\in \mathcal {E}_r(n)$ and $f \tilde {f}\in \mathcal {E}_{2r}(n)\subset \mathcal {E}_{r+1}(n)$ . Suppose that $f^{R}_{N-1}\in \mathcal {E}_r(n)$ satisfies $f f^{R}_{N-1}\in \mathcal {E}_{N-1}(n)$ . By Lemma 3.3, we have $\widetilde {f f^{R}_{N-1}}\in \mathcal {E}_{N-1}(n)$ and $f f^{R}_{N-1} \widetilde {f f^{R}_{N-1}}\in \mathcal {E}_{2N-2}(n)\subset \mathcal {E}_{N}(n)$ . Then we have $f^{R}_N= f^{R}_{N-1}\widetilde {f f^{R}_{N-1}}\in \mathcal {E}_{r}(n)$ and $f f^{R}_N\in \mathcal {E}_{N}(n)$ . Since $\widetilde {f f^{R}_{N-1}}\in \mathcal {E}_{N-1}(n)$ , we have $f^{R}_{N}\sim _{N-1} f^{R}_{N-1}$ . The case for $f^{L}_N$ is similar.

Proposition 3.5. For $N\geq 1$ , we have a filtration of groups

$$ \begin{align*}\mathcal{E}_1(n)/\sim_{N} \;\supset \mathcal{E}_2(n)/\sim_{N} \;\supset \cdots \supset \mathcal{E}_{N-1}(n)/\sim_{N} \;\supset \mathcal{E}_N(n)/\sim_{N}\;=1.\end{align*} $$

Moreover, this is an N-series.

Proof. Firstly, we show that $\mathcal {E}_r(n)/\sim _{N}$ is a group for each $r\geq 1$ . For $f,f',g\in \mathcal {E}_r(n)$ such that $f\sim _N f'$ , we can easily check that $fg\sim _N f'g$ and $gf\sim _N gf'$ . Thus, the composition makes the set $\mathcal {E}_r(n)/\sim _{N}$ a monoid. For $[f]\in \mathcal {E}_r(n)/\sim _{N}$ , by Lemma 3.4, it follows that $[f][f^{R}_N]=[f^{L}_N][f]=1 \in \mathcal {E}_r(n)/\sim _{N}$ . Since $\mathcal {E}_r(n)/\sim _{N}$ is a monoid, we have $[f^{R}_N]=[f^{L}_N]$ , and this is the inverse of $[f]$ . Therefore, $\mathcal {E}_r(n)/\sim _{N}$ is a group for each $r\geq 1$ .

Since $\mathcal {E}_r(n)\supset \mathcal {E}_{r+1}(n)$ , we have $\mathcal {E}_r(n)/\sim _{N} \; \supset \mathcal {E}_{r+1}(n)/\sim _{N}$ . Secondly, we show that the descending series is an N-series. It suffices to show that, for $f\in \mathcal {E}_r(n), g\in \mathcal {E}_s(n)$ , we have

$$ \begin{align*}[[f],[g]]=[f][g][f]^{-1}[g]^{-1}=[fgf^{R}_{N}g^{R}_{N}]\in\mathcal{E}_{r+s}(n)/\sim_{N}. \end{align*} $$

Note that, by Lemma 3.4, we can take $f^{R}_{N}, g^{R}_{N}\in \mathcal {E}_r(n)$ such that $ff^{R}_{N}, gg^{R}_{N}\in \mathcal {E}_{N}(n)\cap \mathcal {E}_{r+s}(n)$ . By commutator calculus, for $x\in F_n$ , we have

$$ \begin{align*}[fg,x]=[f,[g,x]]\;[g,x]\;[f,x]\equiv [g,x]\;[f,x] \quad (\bmod\: \Gamma_{r+s+1}), \end{align*} $$

$$ \begin{align*}[g,[g^{R}_{N},x]\;[f^{R}_{N},x]]=[g,[g^{R}_{N},x]]\; {}^{[g^{R}_{N},x]}[g,[f^{R}_{N},x]]\equiv [g,[g^{R}_{N},x]] \quad (\bmod\: \Gamma_{r+s+1}). \end{align*} $$

Similarly, we have

$$ \begin{align*}[f^{R}_{N} g^{R}_{N},x]\equiv [g^{R}_{N},x]\;[f^{R}_{N},x]\quad (\bmod\: \Gamma_{r+s+1}),\end{align*} $$

$$ \begin{align*}[f,[g^{R}_{N},x]\;[f^{R}_{N},x]]\equiv [f,[f^{R}_{N},x]]\quad (\bmod\: \Gamma_{r+s+1}). \end{align*} $$

Thus, we have

$$ \begin{gather*} \begin{aligned} [fg,[f^{R}_{N} g^{R}_{N},x]]&\equiv [g,[f^{R}_{N} g^{R}_{N},x]]\;[f,[f^{R}_{N} g^{R}_{N},x]]\\ &\equiv [g,[g^{R}_{N},x]\;[f^{R}_{N},x]]\;[f,[g^{R}_{N},x]\;[f^{R}_{N},x]]\\ &\equiv [g,[g^{R}_{N},x]]\;[f,[f^{R}_{N},x]] \quad (\bmod\: \Gamma_{r+s+1}). \end{aligned} \end{gather*} $$

Therefore, we have

$$ \begin{gather*} \begin{aligned} [f g f^{R}_{N} g^{R}_{N},x]&= [fg,[f^{R}_{N} g^{R}_{N},x]]\;[f^{R}_{N} g^{R}_{N},x]\;[fg,x]\\ &\equiv [g,[g^{R}_{N},x]]\;[f,[f^{R}_{N},x]]\;[g^{R}_{N},x]\;[f^{R}_{N},x]\;[g,x]\;[f,x]\\ &\equiv [g,[g^{R}_{N},x]]\;[g^{R}_{N},x]\;[g,x]\;[f,[f^{R}_{N},x]]\;[f^{R}_{N},x]\;[f,x]\\ &=[gg^{R}_{N},x]\;[ff^{R}_{N},x]\\ &\equiv 1 \quad (\bmod\: \Gamma_{r+s+1}), \end{aligned} \end{gather*} $$

and the proof is complete.

For $N\geq r\geq 1$ , we have a canonical projection

$$ \begin{align*}p_{N+1}:\mathcal{E}_r(n)/\sim_{N+1}\twoheadrightarrow\mathcal{E}_r(n)/\sim_{N}.\end{align*} $$

Let $\hat {\mathcal {E}}_r(n)$ denote the projective limit $\underset {N}{\varprojlim } (\mathcal {E}_r(n)/\sim _N)$ and

$$ \begin{align*}\pi_{N}:\hat{\mathcal{E}}_r(n)\twoheadrightarrow\mathcal{E}_r(n)/\sim_N \end{align*} $$

denote the projection. By Proposition 3.5, we have a descending series of groups

$$ \begin{align*}\hat{\mathcal{E}}_1(n)\supset \hat{\mathcal{E}}_2(n)\supset \cdots\end{align*} $$

satisfying

$$ \begin{align*}\bigcap_{r\geq 1} \hat{\mathcal{E}}_r(n)=\{\operatorname{id}\}.\end{align*} $$

Proposition 3.6. The descending series $\hat {\mathcal {E}}_{\ast }(n):=(\hat {\mathcal {E}}_r(n))_{r\geq 1}$ is an N-series.

Proof. By Proposition 3.5, we have $[\mathcal {E}_{r}(n)/\sim _{N},\mathcal {E}_{s}(n)/\sim _{N}]\subset \mathcal {E}_{r+s}(n)/\sim _{N}$ for each $N>r,s$ . By taking the projective limits, we have $[\hat {\mathcal {E}}_{r}(n),\hat {\mathcal {E}}_{s}(n)]\subset \hat {\mathcal {E}}_{r+s}(n)$ .

We have a graded Lie algebra $\mathrm {gr}(\hat {\mathcal {E}}_{\ast }(n))$ associated to the N-series $\hat {\mathcal {E}}_{\ast }(n)$ . Let $\operatorname {gr}^r(\mathcal {E}_{\ast }(n)):=\mathcal {E}_r(n)/\sim _{r+1}$ for $r\geq 1$ and $\mathrm {gr}(\mathcal {E}_{\ast }(n)):=\bigoplus _{r\geq 1}\operatorname {gr}^r(\mathcal {E}_{\ast }(n))$ .

Proposition 3.7. We have a group isomorphism

$$ \begin{align*}\bar{\pi}_{r+1}:\operatorname{gr}^r(\hat{\mathcal{E}}_{\ast}(n))\xrightarrow{\cong}\operatorname{gr}^r(\mathcal{E}_{\ast}(n))\end{align*} $$

induced by the projection $\pi _{r+1}:\hat {\mathcal {E}}_{r}(n)\rightarrow \operatorname {gr}^r(\mathcal {E}_{\ast }(n))$ . Therefore, $\mathrm {gr}(\mathcal {E}_{\ast }(n))$ is a graded Lie algebra.

Proof. The projection $\pi _{r+1}$ induces $\bar {\pi }_{r+1}$ since, for $f\in \hat {\mathcal {E}}_{r+1}(n)$ , we have $\pi _{r+1}(f)\in \mathcal {E}_{r+1}(n)/\sim _{r+1}=1$ .

We will check that $\bar {\pi }_{r+1}$ is surjective. For any $f\in \mathcal {E}_r(n)$ , let $\Phi (f)\in \hat {\mathcal {E}}_r(n)$ satisfy $\pi _N(\Phi (f))=[f]\in \mathcal {E}_r(n)/\sim _{N}$ for each $N>r$ . We have $\bar {\pi }_{r+1}([\Phi (f)])=\pi _{r+1}(\Phi (f))=[f]\in \mathcal {E}_r(n)/\sim _{r+1}$ . Therefore, $\bar {\pi }_{r+1}$ is surjective.

Finally, we show that $\bar {\pi }_{r+1}$ is injective. Let $f\in \hat {\mathcal {E}}_r(n)$ satisfy $\bar {\pi }_{r+1}([f])=1\in \mathcal {E}_r(n)/\sim _{r+1}$ and $\pi _{N}(f)=[f_N]\in \mathcal {E}_r(n)/\sim _{N}$ for $f_N\in \mathcal {E}_r(n)$ . Then, we have $f_{r+1}\in \mathcal {E}_{r+1}(n)$ and $f_N\sim _{r+1}f_{r+1}$ for any $N>r$ . Therefore, we have $\pi _N(f)=[f_N]\in \mathcal {E}_{r+1}(n)/\sim _{N}$ for each $N>r$ and thus $[f]=1\in \operatorname {gr}^r(\hat {\mathcal {E}}_{\ast }(n))$ . The proof is complete.

3.4 Johnson homomorphism of $\operatorname {End}(F_n)$

For $r\geq 1$ , by using Lemma 3.2, we can define a monoid homomorphism

$$ \begin{align*}\tilde{\tau}^{\prime}_{r}:\mathcal{E}_r(n)\rightarrow\operatorname{Hom}(H, \mathcal{L}_{r+1}(n))\end{align*} $$

by $\tilde {\tau }^{\prime }_{r}(f)(x\Gamma _2):=[f,x]\Gamma _{r+2}$ for $f\in \mathcal {E}_r(n), x\in F_n$ . It is easily checked that the monoid homomorphism $\tilde {\tau }^{\prime }_{r}$ induces an injective group homomorphism

$$ \begin{align*}\tilde{\tau}_r:\operatorname{gr}^r(\mathcal{E}_{\ast}(n))\hookrightarrow\operatorname{Hom}(H,\mathcal{L}_{r+1}(n)). \end{align*} $$

We call it the r-th Johnson homomorphism of $\operatorname {End}(F_n)$ .

Proposition 3.8. The map $\tilde {\tau }_r:\operatorname {gr}^r(\mathcal {E}_{\ast }(n))\hookrightarrow \operatorname {Hom}(H, \mathcal {L}_{r+1}(n))$ is an abelian group isomorphism.

Proof. It suffices to show that $\tilde {\tau }_r$ is surjective. For any $\varphi \in \operatorname {Hom}(H,\mathcal {L}_{r+1}(n))$ , we fix a representative of $\varphi (x_i\Gamma _2)\in \mathcal {L}_{r+1}(n)$ and write it $\varphi (x_i)\in \Gamma _{r+1},$ for $i\in [n]$ . Define $\psi \in \operatorname {End}(F_n)$ by

$$ \begin{align*}\psi(x_i)= \varphi(x_i)x_i \text{ for }i\in[n].\end{align*} $$

It turns out that $[\psi ,x]\Gamma _{r+2}=\varphi (x\Gamma _{2})\in \mathcal {L}_{r+1}(n)$ for any $x\in F_n$ by induction on the word length of $x\in F_n$ . Therefore, we have $\tilde {\tau }_r(\psi )=\varphi $ , and thus the map $\tilde {\tau }_r$ is surjective.

Then we obtain the following commutative diagram:

Remark 3.9. It is well known that the Andreadakis filtration $\mathcal {A}_{\ast }(n)$ of $\operatorname {Aut}(F_n)$ includes the lower central series of $\operatorname {IA}(n)$ :

$$ \begin{align*}\Gamma_r(\operatorname{IA}(n))\subset\mathcal{A}_{r}(n). \end{align*} $$

We have $\mathcal {A}_1(n)=\operatorname {IA}(n)$ by definition. Andreadakis [1] conjectured that

(3.6) $$ \begin{align} \mathcal{A}_r(n)=\Gamma_r(\operatorname{IA}(n)) \end{align} $$

for all $r\geq 2,n\geq 2$ . Andreadakis [1] ( $n=3$ ) and Kawazumi [17] (for any n) showed that equation (3.6) holds for $r=2$ . Moreover, Andreadakis [1] showed that the first Johnson homomorphism $\tau _1$ of $\operatorname {Aut}(F_n)$ is an isomorphism. Therefore, we have abelian group isomorphisms

(3.7) $$ \begin{align} \operatorname{gr}^1(\operatorname{IA}(n))\cong \operatorname{Hom}(H,\mathcal{L}_2(n))\cong \operatorname{gr}^1(\mathcal{E}_{\ast}(n)). \end{align} $$

Recently, Satoh [23] showed that equation (3.6) holds for $r=3$ . On the other hand, Bartholdi [5] showed that

$$ \begin{align*}(\mathcal{A}_{5}(3)/ \Gamma_5(\operatorname{IA}(3)))\otimes \mathbb{Q}\cong \mathbb{Q}^{\oplus 3},\end{align*} $$

which is a counterexample of the Andreadakis conjecture. Now, the Andreadakis conjecture remains open for $n\gg r$ .

3.5 The derivation Lie algebra

By equation (3.1) and Proposition 3.8, we have abelian group isomorphisms

$$ \begin{align*}\operatorname{gr}^r(\mathcal{E}_{\ast}(n))\cong H^{\ast}\otimes \mathcal{L}_{r+1}(n)\cong \operatorname{Der}_r(\mathcal{L}_{\ast}(n)) \cong T_r(n).\end{align*} $$

We write $\tilde {\tau }_r:\operatorname {gr}^r(\mathcal {E}_{\ast }(n))\xrightarrow {\cong } \operatorname {Der}_r(\mathcal {L}_{\ast }(n))$ as well.

Proposition 3.10. The abelian group isomorphism

$$ \begin{align*}\tilde{\tau}=\bigoplus_{r\geq 1} \tilde{\tau}_r:\mathrm{gr}(\mathcal{E}_{\ast}(n))\xrightarrow{\cong} \operatorname{Der}(\mathcal{L}_{\ast}(n))\end{align*} $$

is an isomorphism of graded Lie algebras.

Proof. We only need to check that the Lie bracket of $\mathrm {gr}(\mathcal {E}_{\ast }(n))$ is sent to the Lie bracket of $\operatorname {Der}(\mathcal {L}_{\ast }(n))$ . For $f\in \hat {\mathcal {E}}_r(n),g\in \hat {\mathcal {E}}_s(n)$ and $x\in F_n$ , we have

$$ \begin{gather*} \begin{aligned} [\tilde{\tau}_r([f]),\tilde{\tau}_s([g])](x\Gamma_2)&=\tilde{\tau}_r([f])\tilde{\tau}_s([g])(x\Gamma_2)-\tilde{\tau}_s([g])\tilde{\tau}_r([f])(x\Gamma_2)\\ &=[f,[g,x]]\;[g,[f,x]]^{-1}=[[f,g],x]\in \mathcal{L}_{r+s+1}(n). \end{aligned} \end{gather*} $$

On the other hand, we have

$$ \begin{align*}\tilde{\tau}_{r+s}([[f,g]])(x\Gamma_2)=[[f,g],x]\in \mathcal{L}_{r+s+1}(n).\end{align*} $$

Therefore, $\tilde {\tau }$ is an isomorphism of graded Lie algebras.

Remark 3.11. The tree module $\bigoplus _{r\geq 1}T_r(n)$ also has a graded Lie algebra structure which is induced by the Lie algebra structure of $\operatorname {Der}(\mathcal {L}_{\ast }(n))$ . The Lie bracket

$$ \begin{align*}[\cdot,\cdot]:T_r(n)\times T_s(n)\rightarrow T_{r+s}(n)\end{align*} $$

is defined by the difference between two linear sums obtained by contracting the root of one of the trees and the leaves of the other tree

4 Action of $\mathrm {gr}(\mathcal {E}_{\ast }(n))$ on $B_{d}(n)$

We defined the bracket maps (2.2) in [16]. In this section, we extend them to linear maps

$$ \begin{align*}[\cdot,\cdot]:B_{d,k}(n) \otimes \operatorname{gr}^r(\mathcal{E}_{\ast}(n))\rightarrow B_{d,k+r}(n).\end{align*} $$

In Section 4.1, we state Theorem 4.1, which we use to obtain the extended bracket map. In Section 4.2, we extend the category $\mathbf {A}$ to a category $\mathbf {A}^{L}$ , which includes a Lie algebra structure besides the Hopf algebra structure in $\mathbf {A}$ . In Section 4.3, we observe some relations for morphisms of $\mathbf {A}^{L}$ . By using these relations, we prove Theorem 4.1 in Section 4.4.

4.1 Bracket map $[\cdot ,\cdot ]: B_{d,k}(n)\otimes \operatorname {gr}^r(\mathcal {E}_{\ast }(n))\rightarrow B_{d,k+r}(n)$

We have a right $\operatorname {End}(F_n)$ -action on $A_d(n)$ by letting

$$ \begin{align*}u\cdot g:=A_d(g)(u)\end{align*} $$

for $u\in A_d(n), g\in \operatorname {End}(F_n)$ . We define

(4.1) $$ \begin{align} [\cdot,\cdot]:A_d(n)\times\operatorname{End}(F_n)\rightarrow A_d(n) \end{align} $$

by $[u,g]:=u\cdot g -u$ for $u\in A_d(n),g\in \operatorname {End}(F_n)$ , which we call the bracket map.

Theorem 4.1. The N-series $\hat {\mathcal {E}}_{\ast }(n)$ acts on the right on the filtered vector space $A_d(n)$ . That is, we have

$$ \begin{align*}[A_{d,k}(n),\mathcal{E}_r(n)]\subset A_{d,k+r}(n)\end{align*} $$

for any $r\geq 1$ .

Note that we have $[A_{d,k}(n),\Gamma _{r}(\operatorname {IA}(n))]\subset A_{d,k+r}(n)$ (see Lemma 5.7 in [16]). We will prove Theorem 4.1 in Section 4.4.

By using Theorem 4.1, we can extend the bracket map

$$ \begin{align*}[\cdot,\cdot]:B_{d,k}(n)\otimes\operatorname{gr}^r(\operatorname{IA}(n))\rightarrow B_{d,k+r}(n)\end{align*} $$

to $\operatorname {gr}^r(\mathcal {E}_{\ast }(n))$ .

Corollary 4.2. Let $r\geq 1$ . The bracket map (4.1) induces a $\Bbbk $ -linear map

$$ \begin{align*} [\cdot,\cdot]:B_{d,k}(n)\otimes\operatorname{gr}^r(\mathcal{E}_{\ast}(n))\rightarrow B_{d,k+r}(n). \end{align*} $$

We can also extend the $\operatorname {GL}(n;\mathbb {Z})$ -module map

$$ \begin{align*}\beta_{d,k}^r:\operatorname{gr}^r(\operatorname{IA}(n))\rightarrow\operatorname{Hom}(B_{d,k}(n),B_{d,k+r}(n))\end{align*} $$

defined by $\beta _{d,k}^r(g)(u)=[u,g]$ for $g\in \operatorname {gr}^r(\operatorname {IA}(n)),u\in B_{d,k}(n)$ to a group homomorphism

$$ \begin{align*}\tilde{\beta}_{d,k}^r:\operatorname{gr}^r(\mathcal{E}_{\ast}(n))\rightarrow\operatorname{Hom}(B_{d,k}(n),B_{d,k+r}(n)), \end{align*} $$

which $\beta _{d,k}^r$ factors through. That is, we have $\beta _{d,k}^r=\tilde {\beta }_{d,k}^r i$ , where the map $i:\operatorname {gr}^r(\operatorname {IA}(n))\rightarrow \operatorname {gr}^r(\mathcal {E}_{\ast }(n))$ is induced by the inclusion map $\Gamma _{r}(\operatorname {IA}(n))\hookrightarrow \mathcal {E}_{r}(n)$ .

Remark 4.3. The right action of the N-series $\hat {\mathcal {E}}_{\ast }(n)$ on $A_d(n)$ induces an action of the graded Lie algebra $\mathrm {gr}(\mathcal {E}_{\ast }(n))$ on the graded vector space $B_{d}(n)$ :

$$ \begin{align*}\mathrm{gr}(\mathcal{E}_{\ast}(n))\xrightarrow{\cong}\mathrm{gr}(\hat{\mathcal{E}}_{\ast}(n))\rightarrow \bigoplus_{r\geq 1}\operatorname{End}_r(B_{d}(n)),\end{align*} $$

which is given by the group homomorphisms $\tilde {\beta }_{d,k}^r$ . This induced action can be regarded as an action of the derivation Lie algebra $\operatorname {Der}(\mathcal {L}_{\ast }(n))$ on the graded vector space $B_d(n)$ by the identification in Section 3.5.

4.2 The category $\mathbf {A}^{L}$ of extended Jacobi diagrams in handlebodies

The category $\mathbf {A}$ has a cocommutative Hopf algebra with the underlying object $1$ , which we recalled in Section 2.4. Moreover, the morphisms of the category $\mathbf {A}$ have Jacobi diagrams, and the STU relations correspond to relations of Lie algebras. In a proof of Theorem 4.1, we use graphical computations which deal with the Hopf algebra structure and the Lie algebra structure. For this purpose, we extend the category $\mathbf {A}$ to another category $\mathbf {A}^{L}$ which includes the Hopf algebra structure and the Lie algebra structure. In Appendix A, we give an expected presentation of the category $\mathbf {A}^{L}$ .

Construct the category $\mathbf {A}^{L}$ as follows. The set of objects of $\mathbf {A}^{L}$ is the free monoid generated by two objects H and L, where multiplication is denoted by $\otimes $ . The category $\mathbf {A}^{L}$ includes the category $\mathbf {A}$ as a full subcategory with the free monoid generated by H as the set of objects. (On the other hand, the full subcategory with the free monoid generated by L is isomorphic to a category in [13]. See Remark A.4.) In the category $\mathbf {A}^{L}$ , we consider diagrams that are obtained from Jacobi diagrams in handlebodies by attaching univalent vertices of the Jacobi diagrams to the bottom line l and the upper line $l'$ .

Example 4.4. Here is a morphism in $\mathbf {A}^{L}(H\otimes L\otimes H\otimes L\otimes H,H\otimes L^{\otimes 2}\otimes H)$ :

As depicted in Figure 2, the objects H and L in the source of a morphism of $\mathbf {A}^{L}$ correspond to a handle of the handlebody and a univalent vertex attached to the upper line $l'$ , respectively.

Figure 2 Source of a morphism

As depicted in Figure 3, the objects H and L in the target of a morphism of $\mathbf {A}^{L}$ correspond to an arc component mapped into the handlebody and a univalent vertex attached to the bottom line l, respectively.

Figure 3 Target of a morphism

In the category $\mathbf {A}^{L}$ , the object H is considered as a Hopf algebra and L is considered as a Lie algebra. See Section 4.3 and Appendix A.

To define morphisms of the category $\mathbf {A}^{L}$ precisely, we give the following definition.

Definition 4.5. For a finite set T, an $(X_m,T)$ -diagram is a quadruple $(D,V,f,g)$ , where

• • D is a vertex-oriented uni-trivalent graph such that each connected component has at least one univalent vertex,

• • V is a subset of $\partial D=\{\text {univalent vertices of }D\}$ ,

• • f is an embedding of V into the interior of $X_m$ ,

• • g is a bijection from T to $\partial D\setminus V$ .

Note that an $(X_m,\emptyset )$ -diagram is a Jacobi diagram on $X_m$ .

For an object $w=H^{\otimes m_1}\otimes L^{\otimes n_1}\otimes \cdots \otimes H^{\otimes m_r}\otimes L^{\otimes n_r}\in \mathbf {A}^{L}$ , let $m:=\sum _{i=1}^r m_i$ and $n:=\sum _{i=1}^r n_i$ . For $p\geq 0$ , let $[p]^{+}:=\{1^{+},\cdots ,p^{+}\}$ and $[p]^{-}:=\{1^{-},\cdots ,p^{-}\}$ be two copies of $[p]$ .

Definition 4.6. For objects $w=H^{\otimes m_1}\otimes L^{\otimes n_1}\otimes \cdots \otimes H^{\otimes m_r}\otimes L^{\otimes n_r}\in \mathbf {A}^{L}$ and $w'=H^{\otimes m^{\prime }1}\otimes L^{\otimes n^{\prime }1}\otimes \cdots \otimes H^{\otimes m^{\prime }s}\otimes L^{\otimes n^{\prime }s}\in \mathbf {A}^{L}$ , a $(w,w')$ -diagram consists of

• • an $(X_{m'},[n]^{+}\sqcup [n']^{-})$ -diagram $(D,V,f,g)$ such that each connected component of D has at least one univalent vertex in $V\cup g([n']^{-})$

• • a map $\varphi :X_{m'}\cup D\rightarrow U_m$ such that

  1. (1) the pair (the empty set $\emptyset $ , the restriction $\varphi \mid _{X_{m'}}$ ) is an $(m,m')$ -Jacobi diagram; that is, $\varphi $ maps $X_m'$ into $U_m$ in such a way that endpoints of $X_m'$ are arranged in the bottom line l from left to right,

  2. (2) $g([n]^{+})$ is mapped into $l'$ so that the corresponding object in $\mathbf {A}^{L}$ with respect to Figure 2 will be w when we look at the top line $l'$ from left to right,

  3. (3) $g([n']^{-})$ is mapped into l so that the corresponding object in $\mathbf {A}^{L}$ with respect to Figure 3 will be $w'$ when we look at the bottom line l from left to right.

We identify two $(w,w')$ -diagrams if they are homotopic in $U_m$ relative to the endpoints of $X_m'\cup D$ . In what follows, we simply write D for a $(w,w')$ -diagram. For objects w and $w'$ , the hom-set $\mathbf {A}^{L}(w,w')$ is the $\Bbbk $ -vector space spanned by $(w,w')$ -diagrams modulo the STU, AS and IHX relations.

The composition of $\mathbf {A}^{L}$ is defined in a similar way to that of the category $\mathbf {A}$ . We can define a square diagram for an $(w,w')$ -diagram similarly. Let D be a diagram in $\mathbf {A}^{L}(w, w')$ and $D'$ a diagram in $\mathbf {A}^{L}(w', w'')$ . Deform $D'$ to have only the parallel copies of the handle cores in each handle. Then the composition $D'\circ D$ is a diagram obtained by stacking the cabling of D on top of the square presentation of $D'$ .

Example 4.7. For and , the composition $D'\circ D$ is , where the box notation represents a linear sum of Jacobi diagrams. (See [11] and [16] for the definition of the box notation.)

The identity morphism $\operatorname {id}_{H^{\otimes m_1}\otimes L^{\otimes n_1}\otimes \cdots \otimes H^{\otimes m_r}\otimes L^{\otimes n_r}}$ is the following diagram:

We can naturally extend the linear symmetric strict monoidal structure of $\mathbf {A}$ to the category $\mathbf {A}^{L}$ , where the tensor product is defined to be the juxtaposition of the handlebodies.

Note that the symmetries in $\mathbf {A}^{L}$ are determined by

The degree of a $(w,w')$ -diagram is defined by

$$ \begin{align*}\frac{1}{2}\#\{\text{vertices}\}-\#\{\text{univalent vertices attached to the upper line }l'\}.\end{align*} $$

Let $\mathbf {A}^{L}_{d}(w,w')\subset \mathbf {A}^{L}(w,w')$ be the subspace spanned by $(w,w')$ -diagrams of degree d. We have $\mathbf {A}^{L}(w,w')=\bigoplus _{d\geq 0}\mathbf {A}^{L}_{d}(w,w')$ . Since we have

$$ \begin{align*}\mathbf{A}^{L}_{d'}(w',w'')\circ \mathbf{A}^{L}_{d}(w,w')\subset \mathbf{A}^{L}_{d+d'}(w,w'')\end{align*} $$

and

$$ \begin{align*}\mathbf{A}^{L}_{d'}(w,w')\otimes \mathbf{A}^{L}_{d}(z,z')\subset \mathbf{A}^{L}_{d+d'}(w\otimes z,w'\otimes z')\end{align*} $$

for any $w,w',w'',z,z'\in \mathbf {A}^{L}$ , this grading is an $\mathbb {N}$ -grading on $\mathbf {A}^{L}$ . Note that we have $\mathbf {A}_d(m,n)= \mathbf {A}^{L}_d(H^{\otimes m},H^{\otimes n})$ for $m,n\geq 0$ .

4.3 Relations for morphisms in $\mathbf {A}^{L}$

Here, we observe some relations for morphisms of $\mathbf {A}^{L}$ , which we use in the proof of Theorem 4.1.

The cocommutative Hopf algebra $(H,\mu ,\eta ,\Delta ,\epsilon ,S)$ in $\mathbf {A}$ naturally induces a cocommutative Hopf algebra in $\mathbf {A}^{L}$ such that

Additionally, the triple $(L,[\cdot ,\cdot ],c_{L})$ is a Lie algebra with a symmetric invariant $2$ -tensor in $\mathbf {A}^{L}$ (see Appendix A.2), where

Moreover, $\mathbf {A}^{L}$ has two morphisms

The degree of the morphism $c_{L}$ is $1$ and that of the others of the above morphisms is $0$ .

The iterated multiplications

and the iterated comultiplications

for $q\geq 0$ are inductively defined by

$$ \begin{align*}\mu^{[0]}=\eta,\quad\mu^{[1]}=\operatorname{id}_H, \quad \mu^{[q+1]}=\mu(\mu^{[q]}\otimes\operatorname{id}_H) \quad(q\geq 1),\end{align*} $$

$$ \begin{align*}\Delta^{[0]}=\epsilon,\quad\Delta^{[1]}=\operatorname{id}_H, \quad \Delta^{[q+1]}=(\Delta^{[q]}\otimes\operatorname{id}_H)\Delta \quad(q\geq 1).\end{align*} $$

Let

which denotes the adjoint action, and

(4.2)

which denotes the commutator.

Lemma 4.8. We have

1. (1) $S\circ i =-i$

2. (2) $\Delta \circ i=i\otimes \eta +\eta \otimes i$

3. (3) $\epsilon \circ i=0$

4. (4) $ad_{H}(i\otimes i)=-i\circ [\cdot ,\cdot ]$ .

Proof. They can be checked by diagrammatic computation.

Let $\mathfrak {g}$ be a Lie algebra and $U=U(\mathfrak {g})$ be the universal enveloping algebra. We have a filtration $F_{\ast }(U)$ of U induced by the usual filtration of the tensor algebra $T(\mathfrak {g})$ of $\mathfrak {g}$ . Since U has a cocommutative Hopf algebra structure, we can define the commutator operator

$$ \begin{align*}comm:U^{\otimes 2}\rightarrow U \end{align*} $$

in a similar way as equation (4.2). For $x_1,\cdots ,x_m,y_1,\cdots ,y_n\in \mathfrak {g}$ , we have

$$ \begin{align*}comm(x_1\cdots x_m,y_1\cdots y_n)\in F_{\min(m,n)}(U).\end{align*} $$

The following lemma is a diagrammatic version of this fact.

Lemma 4.9.

1. (1) Let $m,n\geq 1$ . We have

   
   where $c_\alpha ,c_\beta \in \mathbb {Z}$ , and where $D_\alpha $ (resp. $D_\beta $ ) is a union of trees with m (resp. n) trivalent vertices. Moreover, for $m=n=1$ , we have
   

2. (2) Let $m\geq 1$ . We have

   

3. (3) We have

   

For example, we have

Proof of Lemma 4.9

By using Lemma 4.8 (2) and

, we have

By Lemma 4.8 (1), it suffices to consider

. By Lemma 4.8 (3), we have

Thus, when $p=0$ , we have $D=0$ . When $p\geq 1$ , by Lemma 4.8 (4), we have

Note that the last term is a $\mathbb {Z}$ -linear sum of unions of tree diagrams with m trivalent vertices. Therefore, the first equality of (1) follows. If $m=n=1$ , then the equality follows from the case where $m=p=1, q=0$ . The second equality of (1) follows similarly.

The first equality of (2) follows from The second equality follows similarly.

We have (3) because

4.4 Proof of Theorem 4.1

In this subsection, we prove Theorem 4.1.

For any $y_1,\cdots ,y_r\in F_n$ , we call $[y_1,\cdots ,[y_{r-1},y_r]]\in \Gamma _r$ an r-fold commutator.

For $i\in [n]$ , define $d_i\in \operatorname {End}(F_n)=\mathbf {F}^{\mathrm {op}}(n,n)$ by

$$ \begin{align*}d_i(x_i)=[y_1,\cdots,[y_{r},y_{r+1}]]^{\epsilon},\quad d_i(x_j)=1 \quad (j\neq i)\end{align*} $$

for $y_1,\cdots ,y_{r+1}\in F_n, \epsilon \in \{\pm 1\}$ , which we call an $(r+1)$ -fold commutator at i. Via the isomorphism $\Bbbk \mathbf {F}^{\mathrm {op}}(n,n)\cong \mathbf {A}_0(n,n)$ , we identify $d_i\in \mathbf {F}^{\mathrm {op}}(n,n)$ with a morphism of the following form

which we also call an $(r+1)$ -fold commutator at i, where each

depicts S or $\operatorname {id}_H$ , and $q_k,p_l\geq 0$ satisfy $\sum _{k=1}^{n}q_k=\sum _{l=1}^{r+1}p_l$ .

Claim 1. An element $g\in \mathcal {E}_r(n)$ can be written as a convolution product

$$ \begin{align*}g= d_{1,1}\ast\cdots\ast d_{1,l_1}\ast\cdots\ast d_{n,1}\ast\cdots\ast d_{n,l_n}\ast\operatorname{id}_{H^{\otimes n}},\end{align*} $$

where $d_{i,j}$ is an $(r+1)$ -fold commutator at i for $i\in [n]$ ( $l_i\geq 0, 1\leq j\leq l_i$ ).

Proof. Let $g\in \mathcal {E}_r(n)$ . Since $\Gamma _{r+1}$ is generated by $(r+1)$ -fold commutators, $g(x_i)x_i^{-1}$ is a product of $(r+1)$ -fold commutators or their inverses for any $i\in [n]$ . Thus, we can decompose g into a convolution product of $(r+1)$ -fold commutators and $\operatorname {id}_{H^{\otimes n}}$ .

Proof of Theorem 4.1

We show that $[A_{d,k}(n),\mathcal {E}_{r}(n)]\subset A_{d,k+r}(n)$ . We can write an element of $A_{d,k}(n)\subset \mathbf {A}^{L}(I,H^{\otimes n})$ as a linear sum of the following diagrams:

where D is a Jacobi diagram with at least k trivalent vertices. Let $g\in \mathcal {E}_{r}(n)$ . By Claim 1, we can write g as a convolution product

$$ \begin{align*}g=d_{1,1}\ast\cdots\ast d_{1,l_1}\ast\cdots\ast d_{n,1}\ast\cdots\ast d_{n,l_n}\ast \operatorname{id}_{H^{\otimes n}},\end{align*} $$

where $d_{i,j}\in \mathbf {A}_0(n,n)$ is an $(r+1)$ -fold commutator at i. Let $l=1+\sum _{i=1}^{n}{l_i}$ .

By using

we have

Here, each

is once connected to all of the diagrams $d_{1,1}, \cdots , d_{n,l_n}$ and $\operatorname {id}_{H^{\otimes n}}$ . Since we have

by Lemma 4.8 (2), the element $u\cdot g$ is a linear sum of diagrams of shape

, where

denotes

or

. If all

that are connected to $\operatorname {id}_{H^{\otimes n}}$ are

, then it is easily checked that the corresponding summand is just u by using Lemma 4.9 (3). Otherwise, at least one of

that are connected to diagrams $d_{1,1},\cdots ,d_{n,l_n}$ are

. By using Lemma 4.9, it follows that each summand is a linear sum of diagrams with at least $k+r$ trivalent vertices. Therefore, we have $[u,g]=u\cdot g-u \in A_{d,k+r}(n)$ .

5 Contraction map

Recall that $H=\mathcal {L}_1(n)= \bigoplus _{i=1}^n \mathbb {Z} \bar {x}_i$ and $H^{\ast }= \bigoplus _{i=1}^n \mathbb {Z} v_i$ . In what follows, we identify $H^{\ast }\otimes \mathcal {L}_{r+1}(n)$ with $T_{r}(n)$ as we remarked in Section 3.2.

5.1 Preliminaries to computation

Let $N\geq 1$ . We briefly review the construction of the irreducible representations of the symmetric group $\mathfrak {S}_N$ . See Fulton–Harris [6] and Sagan [21] for basic facts of representation theory of $\mathfrak {S}_N$ . Let $\lambda =(\lambda _1,\cdots ,\lambda _l)$ be a partition of N, and write $\lambda \vdash N$ . A Young diagram of $\lambda $ consists of $\lambda _i$ boxes in the i-th row for $i\in [l]$ such that the rows of boxes are lined up on the left. A $\lambda $ -tableau is a numbering of the boxes by the integers in $[N]$ . We call a $\lambda $ -tableau standard if the numbering increases in each row and in each column. The canonical $\lambda $ -tableau is a standard tableau whose numbering starts from the first row from left to right and then the second row from left to right and so on.

Let $t_0$ be the canonical $\lambda $ -tableau. Define $R_{t_0}$ (resp. $C_{t_0}$ ) to be the subgroup of $\mathfrak {S}_N$ that preserves each row (resp. column) of $t_0$ . We define

$$ \begin{align*}a_{\lambda}:=\sum_{\sigma\in R_{t_0}}\sigma,\quad b_{\lambda}:=\sum_{\sigma\in C_{t_0}}\mathrm{sgn}(\sigma)\sigma\in\Bbbk\mathfrak{S}_N.\end{align*} $$

For each $\lambda \vdash N$ , the Young symmetrizer $c_{\lambda }$ is defined by

(5.1) $$ \begin{align} c_{\lambda}=b_{\lambda}a_{\lambda}\in\Bbbk\mathfrak{S}_{N}. \end{align} $$

The Specht module $S^{\lambda }$ , which is an irreducible representation of $\mathfrak {S}_N$ corresponding to $\lambda $ , can be constructed as

$$ \begin{align*}S^{\lambda}=\Bbbk\mathfrak{S}_N\cdot c_{\lambda}. \end{align*} $$

Lemma 5.1. We have the following decomposition of $\Bbbk \mathfrak {S}_N$ -bimodules

$$ \begin{gather*} \begin{aligned} \Bbbk\mathfrak{S}_{N}=\bigoplus_{\lambda\vdash N}\Bbbk\mathfrak{S}_{N}\cdot c_{\lambda}\cdot\Bbbk\mathfrak{S}_{N}. \end{aligned} \end{gather*} $$

Proof. This follows from basic facts of representation theory. The reader is referred to [6] and [21].

For $N',N''\geq 0$ , let $N=N'+N''$ . For $\mu \vdash N',\nu \vdash N''$ , let $S^{\mu }\diamond S^{\nu }$ denote the representation of $\mathfrak {S}_N$ induced from the tensor product representation $S^{\mu }\boxtimes S^{\nu }$ of $\mathfrak {S}_{N'}\times \mathfrak {S}_{N''}$ by the inclusion of $\mathfrak {S}_{N'}\times \mathfrak {S}_{N''}$ in $\mathfrak {S}_N$ . By the Littlewood–Richardson rule, we have

$$ \begin{align*}S^{\mu}\diamond S^{\nu}=\bigoplus_{\lambda\vdash N} (S^{\lambda})^{LR_{\mu,\nu}^{\lambda}}, \end{align*} $$

where $LR_{\mu ,\nu }^{\lambda }$ denotes the Littlewood–Richardson coefficient. We have the following lemma by using basic facts of representation theory of $\mathfrak {S}_N$ .

Lemma 5.2. Let $N=N'+N''$ for $N',N''\geq 0$ . Let $\lambda \vdash N,\mu \vdash N',\nu \vdash N''$ , respectively. We have

$$ \begin{align*}\dim_{\Bbbk}((c_{\mu}\diamond c_{\nu})\cdot \Bbbk\mathfrak{S}_N\cdot c_{\lambda})= LR_{\mu,\nu}^{\lambda}.\end{align*} $$

In particular, if the Littlewood–Richardson coefficient $LR_{\mu ,\nu }^{\lambda }=0$ , then we have

$$ \begin{align*}(c_{\mu}\diamond c_{\nu})\cdot \Bbbk\mathfrak{S}_N\cdot c_{\lambda}=0. \end{align*} $$

5.2 Contraction map

We have an isomorphism of $\operatorname {GL}(V_n)$ -modules

(5.2) $$ \begin{align} B_{d,k}(n)\cong V_n^{\otimes 2d-k}\otimes_{\Bbbk\mathfrak{S}_{2d-k}} D_{d,k}, \end{align} $$

where $D_{d,k}$ is the $\Bbbk $ -vector space spanned by $[2d-k]$ -colored open Jacobi diagrams of degree d such that the map $\{\text {univalent vertices of }D\}\rightarrow [2d-k]$ that gives the coloring of D is a bijection. Thus, any element of $B_{d,k}(n)$ can be written in the form

$$ \begin{align*}u(w_1,\cdots,w_{2d-k}):=(w_1\otimes\cdots\otimes w_{2d-k})\otimes u \end{align*} $$

for $u\in D_{d,k}$ and $w_1,\cdots , w_{2d-k}\in V_n$ .

For $\lambda \vdash 2d-k$ , let $B_{d,k}(n)_\lambda $ be the isotypic component of $B_{d,k}(n)$ corresponding to $\lambda $ ; that is,

$$ \begin{align*}B_{d,k}(n)_\lambda\cong V_n^{\otimes 2d-k}\otimes_{\Bbbk\mathfrak{S}_{2d-k}}\Bbbk\mathfrak{S}_{2d-k}c_{\lambda}D_{d,k}.\end{align*} $$

We have $B_{d,k}(n)=\bigoplus _{\lambda \vdash 2d-k}B_{d,k}(n)_{\lambda }$ .

We define a contraction map

$$ \begin{align*}c:B_{d,k}(n)\otimes T_r(n) \rightarrow B_{d,k+r}(n),\end{align*} $$

which is an analogue of the contraction map defined in Appendix B of [6].

Let $p\geq q$ . For $I=(i_1,\cdots ,i_{q})$ such that $i_1,\cdots ,i_{q}$ are distinct elements of $[p]$ , define a contraction map

$$ \begin{align*}c^I:V_n^{\otimes p}\otimes (V_n^{\ast})^{\otimes q}\rightarrow V_n^{\otimes (p-q)} \end{align*} $$

by

$$ \begin{align*}c^I((w_1\otimes\cdots\otimes w_p)\otimes(y_1\otimes\cdots\otimes y_q))= \left(\prod_{j=1}^{q}\langle w_{i_{j}},y_j\rangle\right) w_1\otimes\cdots\hat{w}_{i_1}\cdots\hat{w}_{i_q}\cdots\otimes w_{p}, \end{align*} $$

where $\hat {w}_{i_1}\cdots \hat {w}_{i_q}$ denotes the omission of $w_{i_1},\cdots ,w_{i_q}$ and where $\langle -,-\rangle :V_n\otimes V_n^{\ast }\rightarrow \Bbbk $ denotes the dual pairing. (See [6] for details.)

We next consider a diagrammatic version of the above contraction map $c^I$ . Let $2d-k\geq r+1$ . For $I=(i_1,\cdots ,i_{r+1})\in [2d-k]^{r+1}$ such that $i_1,\cdots ,i_{r+1}$ are distinct, we define a linear map

$$ \begin{align*}c^I:B_{d,k}(n)\otimes T_r(n) \rightarrow B_{d,k+r}(n)\end{align*} $$

by contracting colorings of a Jacobi diagram and leaves of a rooted trivalent tree; that is,

where

We define a contraction map

$$ \begin{align*}c:B_{d,k}(n)\otimes T_r(n) \rightarrow B_{d,k+r}(n) \end{align*} $$

by $c=\sum _{I=(i_1,\cdots ,i_{r+1})\in [2d-k]^{r+1}:\text {distinct}}c^I$ . By using the contraction map c, we define a map

$$ \begin{align*}\gamma_{d,k}^r:T_r(n)\rightarrow \operatorname{Hom}(B_{d,k}(n),B_{d,k+r}(n))\end{align*} $$

by $\gamma _{d,k}^r(g)(u'):=c(u'\otimes g)$ for $g\in T_r(n), u'=u(w_1,\cdots ,w_{2d-k})\in B_{d,k}(n)$ .

5.3 Vanishing conditions for the contraction map

Here, we observe that the contraction map vanishes under certain specific conditions.

For $r\geq 0$ , a trivalent tree is called a based trivalent tree of degree r if it has one distinguished univalent vertex with no coloring (called a base) and $r+1$ univalent vertices (called leaves) that are colored by distinct elements of $[r+1]$ . (Note that a based trivalent tree is different from a rooted trivalent tree.) Let $L_{r}$ denote the $\mathbb {Z}$ -module spanned by based trivalent trees of degree r modulo the AS and IHX relations. The symmetric group $\mathfrak {S}_{r+1}$ acts on the $\mathbb {Z}$ -module $L_{r}$ by the action on colorings of based trivalent trees. Then we have

$$ \begin{align*}\mathcal{L}_{r+1}(n)\cong H^{\otimes (r+1)}\otimes_{\mathbb{Z}\mathfrak{S}_{r+1}} L_{r}.\end{align*} $$

On the other hand, $\mathcal {L}_{r+1}(n)$ has a $\operatorname {GL}(n;\mathbb {Z})$ -module structure by the standard action on each factor. (See [7] for representation theory of $\operatorname {GL}(n;\mathbb {Z})$ .) For $\mu \vdash r+1$ , let $\mathcal {L}_{r+1}(n)_{\mu }$ denote the isotypic component of $\mathcal {L}_{r+1}(n)$ corresponding to $\mu $ ; that is,

$$ \begin{align*}\mathcal{L}_{r+1}(n)_{\mu}\cong H^{\otimes (r+1)}\otimes_{\mathbb{Z}\mathfrak{S}_{r+1}} \mathbb{Z}\mathfrak{S}_{r+1}c_{\mu} L_{r}.\end{align*} $$

We have $\mathcal {L}_{r+1}(n)=\bigoplus _{\mu \vdash r+1}\mathcal {L}_{r+1}(n)_{\mu }$ .

For partitions $\lambda $ and $\mu $ , we write $\lambda \nsupseteq \mu $ if the Young diagram of $\lambda $ does not contain that of $\mu $ .

Proposition 5.3. For $2d-k\geq r+1$ , let $\lambda \vdash 2d-k$ and $\mu \vdash r+1$ . We have

$$ \begin{align*}c(B_{d,k}(n)_{\lambda}\otimes (H^{\ast}\otimes \mathcal{L}_{r+1}(n)_{\mu})) \subset \bigoplus_{\rho:LR^{\lambda}_{\mu,\nu}LR^{\rho}_{\nu,(1)}\neq0 \text{ for some }\nu} B_{d,k+r}(n)_{\rho}. \end{align*} $$

In particular, if $\lambda \nsupseteq \mu $ , then we have

$$ \begin{align*}c(B_{d,k}(n)_{\lambda}\otimes (H^{\ast}\otimes \mathcal{L}_{r+1}(n)_{\mu}))=0.\end{align*} $$

Proof. Any element of $B_{d,k}(n)_{\lambda }$ is a linear sum of $(c_{\lambda }\cdot u)(w_1,\cdots ,w_{2d-k})$ , where $u(w_1,\cdots ,w_{2d-k})\in B_{d,k}(n)$ . Any element of $L_{r}$ is a linear sum of

for $\pi \in \mathfrak {S}_{r+1}$ . Thus, any element of $H^{\ast }\otimes \mathcal {L}_{r+1}(n)_{\mu }$ is a linear sum of $w\otimes ((y_1\otimes \cdots \otimes y_{r+1})\otimes c_{\mu }\cdot L)$ for $w\in H^{\ast }, y_1,\cdots ,y_{r+1}\in H$ .

For any $I=(i_1,\cdots ,i_{r+1})\in [2d-k]^{r+1}$ such that $i_1,\cdots ,i_{r+1}$ are distinct, we have

$$ \begin{align*}c^I((c_{\lambda}\cdot u)(w_1,\cdots,w_{2d-k})\otimes (w\otimes ((y_1 \otimes \cdots\otimes y_{r+1})\otimes c_{\mu}\cdot L))) =\prod_{j=1}^{r+1}\langle w_{i_j}, y_j \rangle D, \end{align*} $$

where

Let $l=2d-k-r-1$ . By Lemma 5.1, we have

$$ \begin{align*}\operatorname{id}_l=\sum_{\nu\vdash l,1\leq i\leq \dim{S^{\nu}}}\tau_{i,1} c_{\nu}\tau_{i,2},\end{align*} $$

where $\tau _{i,1},\tau _{i,2}\in \Bbbk \mathfrak {S}_l$ . Thus, we have

If $LR^{\lambda }_{\mu ,\nu }=0$ for any $\nu \vdash l$ , then we have $D=0$ by Lemma 5.2. Otherwise, since we have

$$ \begin{align*}\operatorname{id}_1\otimes c_{\nu}\in \bigoplus_{\rho\vdash l+1} (S^{\rho})^{LR^{\rho}_{\nu,(1)}}, \end{align*} $$

by the Littlewood–Richardson rule, it follows that

$$ \begin{align*}D\in\bigoplus_{\rho:LR^{\lambda}_{\mu,\nu}LR^{\rho}_{\nu,(1)}\neq0 \text{ for some }\nu}\left( B_{d,k+r}(n)\right)_{\rho}. \end{align*} $$

If $\lambda \nsupseteq \mu $ , then $LR_{\mu ,\nu }^{\lambda }=0$ for any $\nu \vdash l$ . Thus, we have

$$ \begin{align*}c(B_{d,k}(n)_{\lambda}\otimes (H^{\ast}\otimes \mathcal{L}_{r+1}(n)_{\mu}))=0. \end{align*} $$

Remark 5.4. Note that we have $\mathcal {L}_2(n)=\mathcal {L}_2(n)_{(1^2)}$ . Thus, the restriction

$$ \begin{align*}c:B_{d,k}(n)_{\lambda}\otimes (H^{\ast}\otimes \mathcal{L}_2(n)_{(1^2)})\rightarrow B_{d,k+1}(n)_{\rho}\end{align*} $$

of the contraction map vanishes unless $\rho $ can be obtained from $\lambda $ by taking away one box from each of two different rows of $\lambda $ and then by adding one box.

6 Correspondence between the map $\tilde {\beta }_{d,k}^r$ and the map $\gamma _{d,k}^r$

In this section, we prove that the map $\tilde {\beta }_{d,k}^r$ defined in Section 4 can be identified with the map $\gamma _{d,k}^r$ defined in Section 5 via the Johnson homomorphism of $\operatorname {End}(F_n)$ defined in Section 3.

Theorem 6.1. We have $\tilde {\beta }_{d,k}^r=(-1)^r\cdot \gamma _{d,k}^r\circ \tilde {\tau }_r$ . That is, we have the following commutative diagram (up to sign):

Proof. The $\mathbb {Z}$ -module $H^{\ast }\otimes \mathcal {L}_{r+1}(n)$ is spanned by $v_i\otimes [\bar {x}_{i_1},\cdots ,[\bar {x}_{i_r},\bar {x}_{i_{r+1}}]\cdots ]$ for $i, i_1,\cdots ,i_{r+1}\in [n]$ . Define $\phi \in \operatorname {End}(F_n)$ by

$$ \begin{align*}\phi(x_i)=[x_{i_1},\cdots,[x_{i_r},x_{i_{r+1}}]\cdots]\cdot x_i,\quad \phi(x_j)= x_j \:(j\neq i). \end{align*} $$

It is easily checked that $\phi \in \mathcal {E}_{r}(n)$ and that $\tilde {\tau }_r([\phi ]_r)=v_i\otimes [\bar {x}_{i_1},\cdots ,[\bar {x}_{i_r},\bar {x}_{i_{r+1}}]\cdots ]$ , where $[\phi ]_r\in \operatorname {gr}^r(\mathcal {E}_{\ast }(n))$ denotes the image of $\phi $ under the projection.

Any element of $B_{d,k}(n)$ can be written as a linear sum of

, where $1\leq j_1\leq \cdots \leq j_{2d-k}\leq n$ , by arranging the univalent vertices according to the order of indices of the colorings from left to right. We have

where $\tau ^{-1}\in \mathfrak {S}_{2d-k}$ is the $(r+1,2d-k-r-1)$ -shuffle that maps $[r+1]\subset [2d-k]$ to $\{\alpha _l\}$ , and $\sigma \in \mathfrak {S}_{r+1}$ satisfies $\sigma ^{-1}(l)=\tau (\alpha _l)$ for any $l\in [r+1]$ .

Let

, which can be obtained from u by replacing univalent vertices with

and combining solid lines whose corresponding colorings of u are the same. Then $\tilde {u}$ is a lift of u; that is, we have $\theta _{d,n,k}(\tilde {u})=u$ . By the definition of $\tilde {\beta }_{d,k}^r$ , we have

$$ \begin{align*} \tilde{\beta}_{d,k}^r([\phi]_r)(u)=[u,[\phi]_r]=\theta_{d,n,k+r}([\tilde{u},\phi]). \end{align*} $$

We have

where $\rho ^{-1}\in \mathfrak {S}_{n}$ is the $(r+1,n-r-1)$ -shuffle that maps $[r+1]\subset [n]$ to $\{i_1,\cdots ,i_{r+1}\}$ and $\pi \in \mathfrak {S}_{r+1}$ satisfies $\pi ^{-1}(j)=\rho (i_j)$ for any $j\in [r+1]$ . By using Lemma 4.9, we have for $\beta _1,\cdots ,\beta _{r+1}\geq 0$ ,

(6.1)

In the last case, the corresponding term of $[\tilde {u},\phi ]$ is included in $A_{d,k+r+1}(n)$ .

Thus, by equation (6.1) and Lemma 4.8 (2), we have

7 The $\operatorname {GL}(V_n)$ -module structure of $B_d(n)$

In this section, we consider the $\operatorname {GL}(V_n)$ -module structure of $B_d(n)$ and give a decomposition of $B_d(n)$ with respect to connected parts. Moreover, we compute the irreducible decomposition of $B_d(n)$ for $d=3,4,5$ and that of $B_{d,0}(n),B_{d,1}(n)$ for any d. Lastly, we show the surjectivity of the bracket map which we defined in Section 4.

Let $B_{d,k}^c(n)\subset B_{d,k}(n)$ denote the connected part of $B_{d,k}(n)$ , which is spanned by connected $V_n$ -colored open Jacobi diagrams. Let $D_{d,k}^c\subset D_{d,k}$ denote the connected part of $D_{d,k}$ , which is spanned by connected $[2d-k]$ -colored open Jacobi diagrams. We have an isomorphism of $\operatorname {GL}(V_n)$ -modules

$$ \begin{align*} B_{d,k}^c(n)\cong V_n^{\otimes 2d-k}\otimes_{\Bbbk\mathfrak{S}_{2d-k}} D_{d,k}^c, \end{align*} $$

which is the connected version of equation (5.2).

The direct sum $\bigoplus _{d\geq 0}B_d(n)$ has the following coalgebra structure. This is an analogue of the coalgebra structure of the space of open Jacobi diagrams colored by one element [2]. Let $C=\bigcup _{i\in I}C_i$ be a presentation of a diagram $C\in \bigoplus _{d\geq 0}B_d(n)$ as the disjoint union of its connected components. The comultiplication $\Delta $ is defined by

$$ \begin{align*}\Delta(C)=\sum_{J\subset I}\left(\bigcup_{i\in J}C_i\right)\otimes \left(\bigcup_{i\in I\setminus J}C_i\right).\end{align*} $$

Note that the connected part $\bigoplus _{d,k\geq 0}B_{d,k}^c(n)$ coincides with the primitive part of the coalgebra $\bigoplus _{d\geq 0}B_d(n)$ .

7.1 Decomposition of $B_d(n)$ with respect to connected parts

Note that $D_{d,k}^c\neq 0$ if and only if $d-1\leq k\leq 2d-2$ because each element of $D_{d,k}^c$ has at least two univalent vertices and is connected. For $d\geq 1,k\geq 0$ , the pair $(d,k)$ is called a good pair if $d-1\leq k\leq 2d-2$ . We consider the following decomposition of a pair $(d,k)$ to consider the decomposition of an element of $D_{d,k}$ into the connected parts.

Definition 7.1. Let $d,\ k\geq 0$ . A decomposition of $(d,k)$ into good pairs is a sequence of triples of integers

$$ \begin{align*}\pi = ((a_1,d_1, k_1),\cdots, (a_l,d_l, k_l)) \end{align*} $$

such that $(d_i,k_i)$ are good pairs, $a_i\geq 1$ ,

$$ \begin{align*}\sum_{i=1}^l a_i d_i = d, \quad \sum_{i=1}^l a_i k_i = k, \end{align*} $$

and

$$ \begin{align*}\quad(d_1,k_1)>(d_2,k_2)>\cdots>(d_l,k_l) \end{align*} $$

in the lexicographical order.

Let $\Pi (d,k)$ be the set of all decompositions of $(d,k)$ into good pairs.

For example, we have

(7.1) $$ \begin{align} \Pi(4,2)=\{((1,3,2),(1,1,0)),((1,2,2),(2,1,0)),((2,2,1))\}. \end{align} $$

For any diagram $K\in D_{d,k}$ , we can assign a decomposition of $(d,k)$ into good pairs such that $d_i$ and $k_i$ correspond to the degree and the number of trivalent vertices of each connected component of K, respectively, and $a_i$ corresponds to the multiplicity of $(d_i,k_i)$ . We call a coloring of $K=\bigsqcup _{1\leq i\leq l, 1\leq j\leq a_i} K_{i}^{(j)}\in D_{d,k}$ standard if the set of colorings of $K_{i}^{(j)}\in D_{d_i,k_i}^c$ is

$$ \begin{align*}\left\{\sum_{p=1}^{i-1}(2d_p-k_p)a_p +(j-1)(2d_i-k_i)+1 ,\cdots, \sum_{p=1}^{i-1}(2d_p-k_p)a_p +j(2d_i-k_i)\right\}\end{align*} $$

for each $i\in [l],j\in [a_i]$ .

Theorem 7.2. For $d,k,n\geq 0$ , we have an isomorphism of $\operatorname {GL}(V_n)$ -modules

(7.2) $$ \begin{align} B_{d,k}(n) \cong \bigoplus_{\pi =((a_1,d_1, k_1),\cdots,(a_l,d_l, k_l))\in\Pi(d,k)} \left(\bigotimes_{i=1}^l \operatorname{Sym}^{a_i}(B_{d_i,k_i}^c(n))\right). \end{align} $$

To prove this, we need the following proposition.

Proposition 7.3. Let $d,k\geq 0$ . We have an isomorphism of $\mathfrak {S}_{2d-k}$ -modules

(7.3) $$ \begin{align} D_{d,k}\cong \bigoplus_{\pi =((a_1,d_1,k_1),\cdots,(a_l,d_l,k_l))\in\Pi(d,k)} \operatorname{Ind}_{\prod_{i=1}^l(\mathfrak{S}_{2d_i-k_i}\wr\mathfrak{S}_{a_i})}^{\mathfrak{S}_{2d-k}} (\bigotimes_{i=1}^l(D_{d_i,k_i}^c)^{\otimes a_i}), \end{align} $$

where $\mathfrak {S}_{2d_i-k_i}\wr \mathfrak {S}_{a_i}={\mathfrak {S}_{2d_i-k_i}}^{a_i}\rtimes \mathfrak {S}_{a_i}\subset \mathfrak {S}_{(2d_i-k_i)a_i}$ is the wreath product.

For example, we have an isomorphism of $\mathfrak {S}_6$ -modules for $(d,k)=(4,2)$ , which corresponds to equation (7.1),

$$ \begin{align*}D_{4,2}\cong \operatorname{Ind}_{\mathfrak{S}_{4}\times \mathfrak{S}_{2}}^{\mathfrak{S}_{6}} (D_{3,2}^c \otimes D_{1,0}^c) \oplus \operatorname{Ind}_{\mathfrak{S}_{2}\times (\mathfrak{S}_{2}\wr\mathfrak{S}_{2})}^{\mathfrak{S}_{6}} (D_{2,2}^c \otimes (D_{1,0}^c)^{\otimes 2}) \oplus \operatorname{Ind}_{\mathfrak{S}_{3}\wr\mathfrak{S}_{2}}^{\mathfrak{S}_{6}} (D_{2,1}^c)^{\otimes 2}. \end{align*} $$

For example,

and

Via the above isomorphism, the element

corresponds to the element

Proof of Proposition 7.3 Let $D^{\prime }_{d,k}$ denote the right-hand side of equation (7.3).

For any coset $\sigma \in \mathfrak {S}_{2d-k}/\prod _{i=1}^l (\mathfrak {S}_{2d_i-k_i}\wr \mathfrak {S}_{a_i})$ , we fix a representative $\tilde {\sigma }\in \mathfrak {S}_{2d-k}$ of $\sigma $ .

Any element of $D^{\prime }_{d,k}$ can be written uniquely as a linear sum of

$$ \begin{align*}K=\tilde{\sigma}\cdot\bigotimes_{1\leq i\leq l, 1\leq j\leq a_i} K_{i}^{(j)}, \end{align*} $$

where $K_{i}^{(j)}\in D_{d_i,k_i}^c$ . We assign $\bigsqcup _{1\leq i\leq l, 1\leq j\leq a_i}K_{i}^{(j)}$ a standard coloring in $[2d-k]$ according to the order of the colorings in $\bigsqcup _{i=1}^{l} [2d_i-k_i]^{a_i}$ of $\bigotimes _{1\leq i\leq l, 1\leq j\leq a_i} K_{i}^{(j)}$ . For example, if

then the corresponding coloring of $\bigsqcup _{1\leq i\leq l, 1\leq j\leq a_i}K_{i}^{(j)}$ is

Define a map $\Psi :D^{\prime }_{d,k}\rightarrow D_{d,k}$ by

$$ \begin{align*}\Psi(K)=\tilde{\sigma}\cdot \bigsqcup_{1\leq i\leq l, 1\leq j\leq a_i} K_{i}^{(j)}, \end{align*} $$

where $\tilde {\sigma }\in \mathfrak {S}_{2d-k}$ acts on the colorings in $[2d-k]$ . We can check that the map $\Psi $ is an $\mathfrak {S}_{2d-k}$ -module map.

We need to check that $\Psi $ is bijective. If we have $\Psi (K)=\Psi (L)$ for $K=\tilde {\sigma }\cdot \bigotimes _{1\leq i\leq l, 1\leq j\leq a_i} K_{i}^{(j)}$ , $L=\tilde {\tau }\cdot \bigotimes _{1\leq i\leq l, 1\leq j\leq a_i} L_{i}^{(j)}$ , then we have $\sigma =\tau $ by looking at the set of colorings of each connected component. Since we fix the representatives of cosets of $\mathfrak {S}_{2d-k}/\prod _{i=1}^l (\mathfrak {S}_{2d_i-k_i}\wr \mathfrak {S}_{a_i})$ , we have $\tilde {\sigma }=\tilde {\tau }$ . Thus, we have $K=L$ and $\Psi $ is injective. For any element $K\in D_{d,k}$ , we can take $\sigma \in \mathfrak {S}_{2d-k}/\prod _{i=1}^l (\mathfrak {S}_{2d_i-k_i}\wr \mathfrak {S}_{a_i})$ such that $K=\tilde {\sigma }\cdot \bigsqcup _{1\leq i\leq l, 1\leq j\leq a_i} K_{i}^{(j)}$ , where $K_{i}^{(j)}\in D_{(d_i,k_i)}^c$ and $\bigsqcup _{1\leq i\leq l, 1\leq j\leq a_i} K_{i}^{(j)}$ has a standard coloring. Therefore, $\Psi $ is surjective.

Proof of Theorem 7.2 By Proposition 7.3, we have

$$ \begin{gather*} \begin{aligned} B_{d,k}(n) &\cong V_n^{\otimes 2d-k}\otimes_{\Bbbk\mathfrak{S}_{2d-k}}D_{d,k} \\ &\cong\bigoplus_{\pi\in \Pi(d,k)} \left(V_n^{\otimes 2d-k}\otimes_{\Bbbk\mathfrak{S}_{2d-k}} \operatorname{Ind}_{\prod_{i=1}^l (\mathfrak{S}_{2d_i-k_i}\wr\mathfrak{S}_{a_i})}^{\mathfrak{S}_{2d-k}} \left(\bigotimes_{i=1}^l(D_{d_i,k_i}^c)^{\otimes a_i} \right) \right). \end{aligned} \end{gather*} $$

Moreover, we can check equation (7.2) as follows.

$$ \begin{gather*} \begin{aligned} &V_n^{\otimes 2d-k}\otimes_{\Bbbk\mathfrak{S}_{2d-k}} \operatorname{Ind}_{\prod_{i=1}^l (\mathfrak{S}_{2d_i-k_i}\wr\mathfrak{S}_{a_i})}^{\mathfrak{S}_{2d-k}} \left(\bigotimes_{i=1}^l(D_{d_i,k_i}^c)^{\otimes a_i} \right)\\ &\cong V_n^{\otimes 2d-k}\otimes_{\Bbbk\mathfrak{S}_{2d-k}} \operatorname{Ind}_{\prod_{i=1}^l \mathfrak{S}_{a_i(2d_i-k_i)}}^{\mathfrak{S}_{2d-k}} \left( \operatorname{Ind}_{\prod_{i=1}^l (\mathfrak{S}_{2d_i-k_i}\wr\mathfrak{S}_{a_i})}^{\prod_{i=1}^l \mathfrak{S}_{a_i(2d_i-k_i)}} \left(\bigotimes_{i=1}^l(D_{d_i,k_i}^c)^{\otimes a_i} \right) \right)\\ &\cong V_n^{\otimes 2d-k}\otimes_{\Bbbk\mathfrak{S}_{2d-k}} \operatorname{Ind}_{\prod_{i=1}^l \mathfrak{S}_{a_i(2d_i-k_i)}}^{\mathfrak{S}_{2d-k}} \left( \bigotimes_{i=1}^l \operatorname{Ind}_{\mathfrak{S}_{2d_i-k_i}\wr\mathfrak{S}_{a_i}}^{\mathfrak{S}_{a_i(2d_i-k_i)}} \left((D_{d_i,k_i}^c)^{\otimes a_i} \right) \right)\\ &\cong V_n^{\otimes 2d-k}\otimes_{\Bbbk(\prod_{i=1}^l \mathfrak{S}_{a_i(2d_i-k_i)})} \left( \bigotimes_{i=1}^l \operatorname{Ind}_{\mathfrak{S}_{2d_i-k_i}\wr\mathfrak{S}_{a_i}}^{\mathfrak{S}_{a_i(2d_i-k_i)}} \left((D_{d_i,k_i}^c)^{\otimes a_i} \right) \right)\\ &\cong \bigotimes_{i=1}^l \left( V_n^{\otimes a_i(2d_i-k_i)}\otimes_{\Bbbk\mathfrak{S}_{a_i(2d_i-k_i)}} \left( \operatorname{Ind}_{\mathfrak{S}_{2d_i-k_i}\wr\mathfrak{S}_{a_i}}^{\mathfrak{S}_{a_i(2d_i-k_i)}} \left((D_{d_i,k_i}^c)^{\otimes a_i} \right) \right) \right)\\ &\cong \bigotimes_{i=1}^l \left( V_n^{\otimes a_i(2d_i-k_i)}\otimes_{\Bbbk(\mathfrak{S}_{2d_i-k_i}\wr\mathfrak{S}_{a_i})} \left((D_{d_i,k_i}^c)^{\otimes a_i} \right) \right)\\ &\cong \bigotimes_{i=1}^l \operatorname{Sym}^{a_i}\left( V_n^{\otimes(2d_i-k_i)}\otimes_{\Bbbk\mathfrak{S}_{2d_i-k_i}} D_{d_i,k_i}^c \right)\\ &\cong \bigotimes_{i=1}^l \operatorname{Sym}^{a_i}\left(B_{d_i,k_i}^c(n)\right). \end{aligned} \end{gather*} $$

7.2 Irreducible decomposition of $B_d(n)$ as $\operatorname {GL}(V_n)$ -modules

In this subsection, for simplicity, we write $V=V_n$ , $B_{d,k}=B_{d,k}(n)$ and $B_{d,k}^c=B_{d,k}^c(n)$ .

Let N be a nonnegative integer and $\lambda \vdash N$ . Recall from Section 5.1 that $S^\lambda $ denotes the Specht module, which is an irreducible representation of $\mathfrak {S}_N$ corresponding to $\lambda $ . Let $V_{\lambda }=\mathbb {S}_{\lambda }V$ denote the image of V under the Schur functor $\mathbb {S}_{\lambda }$ . Note that $V_{\lambda }$ is a simple $\operatorname {GL}(V)$ -module if $n\geq r(\lambda )$ and that $V_{\lambda }=0$ if $n<r(\lambda )$ , where $r(\lambda )$ is the number of rows of $\lambda $ .

We use the Littlewood–Richardson rule, plethysms and results by Bar-Natan [4] to compute the irreducible decompositions of the $\operatorname {GL}(V)$ -modules $B_d$ .

Proposition 7.4 Bar-Natan [4]

As $\mathfrak {S}_{2d-k}$ -modules, we have isomorphisms

$$ \begin{align*}D^c_{1,0}\cong S^{(2)},\end{align*} $$

$$ \begin{align*}D^c_{2,1}\cong S^{(1^3)},\quad D^c_{2,2}\cong S^{(2)},\end{align*} $$

$$ \begin{align*}D^c_{3,2}\cong S^{(2^2)},\quad D^c_{3,3}\cong S^{(1^3)},\quad D^c_{3,4}\cong S^{(2)},\end{align*} $$

$$ \begin{align*}D^c_{4,3}\cong S^{(3,1^2)},\quad D^c_{4,4}\cong S^{(4)}\oplus S^{(2^2)} ,\quad D^c_{4,5}\cong S^{(1^3)} ,\quad D^c_{4,6}\cong S^{(2)},\end{align*} $$

$$ \begin{align*}D^c_{5,4}\cong S^{(4,2)}\oplus S^{(2^3)}\oplus S^{(3,1^3)},\quad D^c_{5,5}\cong ({S^{(3,1^2)}})^{\oplus 2},\end{align*} $$

$$ \begin{align*}D^c_{5,6}\cong S^{(4)}\oplus ({S^{(2^2)}})^{\oplus 2},\quad D^c_{5,7}\cong ({S^{(1^3)}})^{\oplus 2},\quad D^c_{5,8}\cong ({S^{(2)}})^{\oplus 2}.\end{align*} $$

Lemma 7.5. We have the following isomorphisms of the $\operatorname {GL}(V)$ -modules:

$$ \begin{align*}B^c_{1,0}\cong V_{(2)},\end{align*} $$

$$ \begin{align*}B^c_{2,1}\cong V_{(1^3)},\quad B^c_{2,2}\cong V_{(2)},\end{align*} $$

$$ \begin{align*}B^c_{3,2}\cong V_{(2^2)},\quad B^c_{3,3}\cong V_{(1^3)},\quad B^c_{3,4}\cong V_{(2)},\end{align*} $$

$$ \begin{align*}B^c_{4,3}\cong V_{(3,1^2)},\quad B^c_{4,4}\cong V_{(4)}\oplus V_{(2^2)} ,\quad B^c_{4,5}\cong V_{(1^3)},\quad B^c_{4,6}\cong V_{(2)},\end{align*} $$

$$ \begin{align*}B^c_{5,4}\cong V_{(4,2)}\oplus V_{(2^3)}\oplus V_{(3,1^3)},\quad B^c_{5,5}\cong (V_{(3,1^2)})^{\oplus 2},\end{align*} $$

$$ \begin{align*}B^c_{5,6}\cong V_{(4)}\oplus (V_{(2^2)})^{\oplus 2},\quad B^c_{5,7}\cong (V_{(1^3)})^{\oplus 2},\quad B^c_{5,8}\cong (V_{(2)})^{\oplus 2}.\end{align*} $$

Proof. These follow from Proposition 7.4.

Proposition 7.6. For $d=3,4,5$ , we have the following irreducible decompositions of the $\operatorname {GL}(V)$ -modules $B_d$ .

1. (1) We have $B_3=B_{3,0}\oplus \cdots \oplus B_{3,4},$ where

   $$ \begin{gather*} \begin{aligned} B_{3,0}&\cong V_{(6)}\oplus V_{(4,2)}\oplus V_{(2^3)},\\ B_{3,1}&\cong V_{(3,1^2)}\oplus V_{(2,1^3)},\\ B_{3,2}&\cong V_{(4)}\oplus V_{(3,1)}\oplus (V_{(2^2)})^{\oplus 2},\\ B_{3,3}&=B^c_{3,3}\cong V_{(1^3)},\\ B_{3,4}&=B^c_{3,4}\cong V_{(2)}. \end{aligned} \end{gather*} $$

2. (2) We have $B_4=B_{4,0}\oplus \cdots \oplus B_{4,6},$ where

   $$ \begin{gather*} \begin{aligned} B_{4,0}&\cong V_{(8)}\oplus V_{(6,2)}\oplus V_{(4^2)}\oplus V_{(4,2^2)}\oplus V_{(2^4)},\\ B_{4,1}&\cong V_{(5,1^2)}\oplus V_{(4,1^3)}\oplus V_{(3^2,1)} \oplus V_{(3,2,1^2)}\oplus V_{(2^2,1^3)},\\ B_{4,2}&\cong V_{(6)}\oplus V_{(5,1)}\oplus(V_{(4,2)})^{\oplus 3}\oplus(V_{(3,2,1)})^{\oplus 2}\oplus(V_{(2^3)})^{\oplus 3}\oplus V_{(2,1^4)},\\ B_{4,3}&\cong(V_{(3,1^2)})^{\oplus 3}\oplus(V_{(2,1^3)})^{\oplus 2},\\ B_{4,4}&\cong(V_{(4)})^{\oplus 3}\oplus V_{(3,1)}\oplus(V_{(2^2)})^{\oplus 3},\\ B_{4,5}&\cong V_{(1^3)},\\ B_{4,6}&\cong V_{(2)}. \end{aligned} \end{gather*} $$

3. (3) We have $B_5=B_{5,0}\oplus \cdots \oplus B_{5,8},$ where

   $$ \begin{gather*} \begin{aligned} B_{5,0}&\cong V_{(10)}\oplus V_{(8,2)}\oplus V_{(6,4)}\oplus V_{(6,2^2)}\oplus V_{(4^2,2)}\oplus V_{(4,2^3)} \oplus V_{(2^5)},\\ B_{5,1}&\cong V_{(7,1^2)}\oplus V_{(6,1^3)}\oplus V_{(5,3,1)}\oplus V_{(5,2,1^2)}\oplus V_{(4,3,1^2)}\oplus V_{(4,2,1^3)}\\ &\oplus V_{(3^3)}\oplus V_{(3^2,2,1)}\oplus V_{(3,2^2,1^2)}\oplus V_{(2^3,1^3)},\\ B_{5,2}&\cong V_{(8)}\oplus V_{(7,1)}\oplus(V_{(6,2)})^{\oplus 3}\oplus V_{(5,3)}\oplus(V_{(5,2,1)})^{\oplus 2}\oplus(V_{(4^2)})^{\oplus 2}\\ &\oplus(V_{(4,3,1)})^{\oplus 2}\oplus(V_{(4,2^2)})^{\oplus 5}\oplus V_{(4,1^4)}\oplus V_{(3^2,1^2)}\oplus(V_{(3,2^2,1)})^{\oplus 3}\\ &\oplus V_{(3,2,1^3)}\oplus V_{(3,1^5)}\oplus(V_{(2^4)})^{\oplus 3}\oplus V_{(2^2,1^4)},\\ B_{5,3}&\cong (V_{(5,1^2)})^{\oplus 3}\oplus(V_{(4,2,1)})^{\oplus 2}\oplus(V_{(4,1^3)})^{\oplus 4}\oplus(V_{(3^2,1)})^{\oplus 4}\oplus(V_{(3,2,1^2)})^{\oplus 5}\\ &\oplus V_{(3,1^4)}\oplus(V_{(2^2,1^3)})^{\oplus 3},\\ B_{5,4}&\cong (V_{(6)})^{\oplus 3}\oplus(V_{(5,1)})^{\oplus 3}\oplus(V_{(4,2)})^{\oplus 8}\oplus(V_{(3,2,1)})^{\oplus 4}\oplus V_{(3,1^3)}\oplus(V_{(2^3)})^{\oplus 6}\\ &\oplus V_{(2^2,1^2)}\oplus V_{(2,1^4)}\oplus V_{(1^6)},\\ B_{5,5}&\cong (V_{(3,1^2)})^{\oplus 5}\oplus(V_{(2,1^3)})^{\oplus 3},\\ B_{5,6}&\cong (V_{(4)})^{\oplus 3}\oplus(V_{(3,1)})^{\oplus 2}\oplus(V_{(2^2)})^{\oplus 4},\\ B_{5,7}&\cong (V_{(1^3)})^{\oplus 2},\\ B_{5,8}&\cong (V_{(2)})^{\oplus 2}. \end{aligned} \end{gather*} $$

Proof. By using Theorem 7.2, Lemma 7.5 and plethysm, we have

$$ \begin{align*}B_{3,0}\cong\operatorname{Sym}^3(B^c_{1,0})\cong\mathbb{S}_{(3)}(\mathbb{S}_{(2)} V) \cong V_{(6)}\oplus V_{(4,2)}\oplus V_{(2^3)}. \end{align*} $$

By using Theorem 7.2, Lemma 7.5 and the Littlewood–Richardson rule, we have

$$ \begin{align*}B_{3,1}\cong B^c_{2,1}\otimes B^c_{1,0}\cong V_{(1^3)}\otimes V_{(2)}\cong V_{(3,1^2)}\oplus V_{(2,1^3)}, \end{align*} $$

and

$$ \begin{align*}B_{3,2}\cong B^c_{3,2}\oplus(B^c_{2,2}\otimes B^c_{1,0})\cong V_{(2^2)}\oplus (V_{(4)}\oplus V_{(3,1)} \oplus V_{(2^2)}). \end{align*} $$

The other isomorphisms of (1) follow from Lemma 7.5.

The irreducible decompositions (2) and (3) follow in a similar way.

We need the irreducible decompositions of $B_{d,0}$ and $B_{d,1}$ to study the $\operatorname {Aut}(F_n)$ -module structure of $A_d(n)$ . For $\lambda =(\lambda _1,\cdots ,\lambda _r)\vdash N$ , let $2\lambda $ denote the partition $(2\lambda _1,\cdots ,2\lambda _r)$ of $2N$ .

Proposition 7.7. For any $d\geq 0$ , we have

$$ \begin{align*}B_{d,0}\cong \bigoplus_{\lambda\vdash d} V_{2\lambda}. \end{align*} $$

For any $d\geq 2$ , we have

$$ \begin{align*}B_{d,1}\cong \bigoplus_{\lambda\vdash 2d-1 \text{ with exactly }3\text{ odd parts}}V_{\lambda}. \end{align*} $$

Proof. By Theorem 5.4.23 in [14], we have

$$ \begin{align*}\mathbb{S}_{(d)}(\mathbb{S}_{(2)} V)\cong\bigoplus_{\lambda\vdash d} V_{2\lambda}.\end{align*} $$

Therefore, by Theorem 7.2 and Lemma 7.5, we have

$$ \begin{align*}B_{d,0}\cong\operatorname{Sym}^{d}(B^c_{1,0})\cong\mathbb{S}_{(d)}(\mathbb{S}_{(2)} V)\cong \bigoplus_{\lambda\vdash d} V_{2\lambda}.\end{align*} $$

By Theorem 7.2, Lemma 7.5, plethysm and the Littlewood–Richardson rule, we have

$$ \begin{align*}B_{d,1}\cong B^c_{2,1}\otimes \operatorname{Sym}^{d-2}(B^c_{1,0})\cong V_{(1^3)}\otimes \bigoplus_{\mu\vdash d-2} V_{2\mu} \cong \bigoplus_{\lambda\vdash 2d-1\text{ with exactly }3\text{ odd parts}}V_{\lambda}.\end{align*} $$

7.3 Surjectivity of the bracket map $[\cdot ,\cdot ]:B_{d,k}(n)\otimes \operatorname {gr}^1(\operatorname {IA}(n))\rightarrow B_{d,k+1}(n)$

Here, we show that the bracket map $[\cdot ,\cdot ]: B_{d,k}(n)\otimes \operatorname {gr}^1(\operatorname {IA}(n))\rightarrow B_{d,k+1}(n)$ is surjective for $n\geq 2d$ . Since we have abelian group isomorphisms (3.7), the bracket map of $\operatorname {gr}^1(\operatorname {IA}(n))$ coincides with that of $\operatorname {gr}^1(\mathcal {E}_{\ast }(n))$ . Thus, we can compute the bracket map by using the contraction map c defined in Section 5.

Define $K_{i,j}, K_{i,j,k}\in \operatorname {IA}(n)$ by

$$ \begin{align*}K_{i,j}(x_i)=x_jx_ix_j^{-1},\quad K_{i,j}(x_l)=x_l \quad(l\neq i),\end{align*} $$

(7.4) $$ \begin{align} K_{i,j,k}(x_i)=x_i[x_j,x_k],\quad K_{i,j,k}(x_l)=x_l \quad(l\neq i). \end{align} $$

Proposition 7.8. For $n\geq 2d-k$ , the bracket map

$$ \begin{align*}[\cdot,\cdot]:B_{d,k}(n)\otimes \operatorname{gr}^1(\operatorname{IA}(n))\rightarrow B_{d,k+1}(n)\end{align*} $$

is surjective.

Proof. Any element of $B_{d,k+1}(n)$ is a linear sum of , where $i_1,\cdots ,i_{2d-k-1}\in [n]$ . Since $n\geq 2d-k$ , we can take , where $i,j\in [n]\setminus \{i_2,\cdots ,i_{2d-k-1}\}$ are distinct. We have $[\tilde {u},K_{i_1,j,i}]=u$ , and therefore, the bracket map is surjective.

As in Section 5.3, for $\lambda \vdash 2d-k$ , let $B_{d,k}(n)_{\lambda }$ denote the isotypic component of $\operatorname {GL}(n;\mathbb {Z})$ -module $B_{d,k}(n)$ corresponding to $\lambda $ .

In Proposition 7.7, we computed a decomposition of $B_{d,0}(n)$ . Since the Young diagram of $(2d)$ does not contain that of $(1^2)$ , by Remark 5.4, we have the following corollary.

Corollary 7.9. The restriction of the bracket map

$$ \begin{align*}[\cdot,\cdot]:\bigoplus_{\lambda\vdash d,\lambda\neq (d)} B_{d,0}(n)_{2\lambda}\otimes \operatorname{gr}^1(\operatorname{IA}(n))\rightarrow B_{d,1}(n)\end{align*} $$

is surjective for $n\geq 2d$ .

Lastly, we consider the condition for $\lambda \vdash 2d-k$ that the isotypic component $B_{d,k}(n)_{\lambda }$ of $B_{d,k}(n)$ does not vanish. Let $o(\lambda )$ be the number of odd parts of $\lambda $ . We have

$$ \begin{align*}o(\lambda)\equiv 2d-k\equiv k \quad(\bmod\: 2).\end{align*} $$

In Proposition 7.7, we observed that $o(\lambda )=0\; (k=0)$ and $o(\lambda )=3 \;(k=1)$ . Moreover, by Proposition 7.8 and Remark 5.4, we have $o(\lambda )\leq 3k$ if $B_{d,k}(n)_{\lambda }\neq 0$ .

8 The $\operatorname {Aut}(F_n)$ -module structure of $A_d(n)$

In this section, we study the $\operatorname {Aut}(F_n)$ -module structure of $A_d(n)$ . We have $A_0(n)=\Bbbk $ for any $n\geq 0$ , and we studied the cases where $d=1, 2$ in [16]. Note that we have $A_d(0)=0$ for $d\geq 1$ . Thus, we have only to consider $n\geq 1$ . Here, we construct a direct decomposition of $A_d(n)$ as $\operatorname {Aut}(F_n)$ -modules for any $d\geq 3,n\geq 1$ , which is indecomposable for $n\geq 2d$ . Moreover, we study the degree $3$ case in detail.

8.1 A direct decomposition of $A_d(n)$

Here, we give a direct decomposition of the $\operatorname {Aut}(F_n)$ -module $A_d(n)$ .

Let

, and depict it as

Here, we use the same graphical notation of morphisms $\mu ,\eta ,\Delta ,\epsilon ,S$ in the category $\mathbf {A}$ as in the category $\mathbf {A}^{L}$ . As in Section 4.3, we can define the iterated multiplications $\mu ^{[q]}\in \mathbf {A}(q,1)$ for $q\geq 0$ . For $m\geq 0$ , there is a group homomorphism

$$ \begin{align*}\mathfrak{S}_{m}\rightarrow \mathbf{A}(m,m),\quad \sigma \mapsto P_{\sigma}, \end{align*} $$

where $P_{\sigma }$ is the symmetry in $\mathbf {A}$ corresponding to $\sigma $ . Set

By Habiro–Massuyeau [11, Lemma 5.16], every element of $A_d(n)$ is a linear combination of morphisms of the form

for $\sigma \in \mathfrak {S}_{2d}$ and $q_1,\cdots , q_n\geq 0$ such that $q_1+\cdots +q_n=2d$ . The following lemma easily follows.

Lemma 8.1. For $n\geq 0$ , we have

$$ \begin{align*}A_d(n)=\operatorname{Span}_\Bbbk\{A_d(f)(c^{\otimes d})\mid f\in\mathbf{F}^{\mathrm{op}}(2d,n)\}. \end{align*} $$

For $X\in A_d(m)$ , let

$$ \begin{align*}A_d X:\mathbf{F}^{\mathrm{op}}\rightarrow \mathbf{fVect}\end{align*} $$

denote the subfunctor of $A_d$ generated by X. That is, for any $n\in \mathbb {N}$ , $A_d X(n)$ is the $\operatorname {Aut}(F_n)$ -submodule of $A_d(n)$ defined by

$$ \begin{align*}A_d X (n):=\operatorname{Span}_\Bbbk\{A_d(f)(X)\mid f\in\mathbf{F}^{\mathrm{op}}(m,n)\}.\end{align*} $$

Set

Note that we have $A_1 Q=0$ .

Theorem 8.2. We have

(8.1) $$ \begin{align} A_d(n)=A_d P(n)\oplus A_d Q(n). \end{align} $$

Proof. By Lemma 8.1, any element of $A_d(n)$ is a linear sum of $A_d(f)(c^{\otimes d})$ for $f\in \mathbf {F}^{\mathrm {op}}(2d,n)$ . Define an $\operatorname {Aut}(F_n)$ -module map

$$ \begin{align*}e_n:A_d(n)\rightarrow A_d(n)\end{align*} $$

by $e_n(A_d(f)(c^{\otimes d}))=\frac {1}{(2d)!} A_d(f)(P)$ for $f\in \mathbf {F}^{\mathrm {op}}(2d,n)$ . This is well defined because the $4$ T relation is sent to $0$ . Since $A_d P$ is generated by P, we have $\operatorname {im}(e_n)=A_d P(n)$ .

Since we have $e_n(A_d(f)(P))=A_d(f)(P)$ for any $f\in \mathbf {F}^{\mathrm {op}}(2d,n)$ , the $\operatorname {Aut}(F_n)$ -endomorphism $e_n$ is an idempotent in $\operatorname {End}(A_d(n))$ , where we consider $A_d(n)$ as a right $\operatorname {Aut}(F_n)$ -module. Therefore, we have

$$ \begin{align*}A_d(n)= \operatorname{im}(e_n)\oplus \ker(e_n),\quad \ker(e_n)=\operatorname{im}(1-e_n).\end{align*} $$

Since , we have $A_d Q(n)\subset \ker (e_n)$ . Finally, we need to check that $\operatorname {im}(1-e_n)\subset A_d Q(n)$ . Since we have for $f\in \mathbf {F}^{\mathrm {op}}(2d,n)$ ,

$$ \begin{gather*} \begin{aligned} (1-e_n)(A_d(f)(c^{\otimes d}))&= A_d(f)(c^{\otimes d})-\frac{1}{(2d)!} A_d(f)(P)\\ &=\frac{1}{(2d)!}\sum_{\sigma\in\mathfrak{S}_{2d}}A_d(f)(c^{\otimes d}- \sigma c^{\otimes d}), \end{aligned} \end{gather*} $$

we need to show that, for any $\sigma \in \mathfrak {S}_{2d}$ , there exists $\tau \in \Bbbk \mathfrak {S}_{2d}$ such that

(8.2) $$ \begin{align} c^{\otimes d}- \sigma c^{\otimes d}=\tau Q\in A_d Q(2d). \end{align} $$

It suffices to show the existence of $\tau $ satisfying equation (8.2) when $\sigma $ is an adjacent transposition because any permutation is generated by adjacent transpositions, and we have such $\tau $ by inductively using

$$ \begin{align*}c^{\otimes d}- \sigma\rho c^{\otimes d}=c^{\otimes d}-\sigma c^{\otimes d}+ \sigma(c^{\otimes d}-\rho c^{\otimes d}).\end{align*} $$

If $\sigma $ is an adjacent transposition $(2i,2i+1)$ for $i\in [n-1]$ , then we set

$$ \begin{align*}\tau= \begin{pmatrix} 1&2&3&4&5&\cdots&2d\\ 2i-1&2i+2&2i+1&2i&1&\cdots \widehat{2i-1}\cdots \widehat{2i+2}\cdots&2d\\ \end{pmatrix}.\end{align*} $$

If $\sigma $ is an adjacent transposition $(2i-1,2i)$ for $i\in [n]$ , then we set $\tau =0$ . The proof is complete.

Lemma 8.3. The $\operatorname {Aut}(F_n)$ -module $A_d P(n)$ is irreducible and thus indecomposable.

Proof. Since , we have $\theta _{d,n}(A_d P(n))=B_{d,0}(n)_{(2d)}$ by the PBW map. Therefore, $A_d P(n)$ is an irreducible $\operatorname {Aut}(F_n)$ -module.

For $\lambda \vdash d$ , set , where $c_{2\lambda }\in \Bbbk \mathfrak {S}_{2d}$ is the Young symmetrizer. Note that we have $Q_{(d)}=P$ .

Lemma 8.4. For $\lambda \vdash d, \lambda \neq (d)$ , we have $Q_{\lambda }\in A_d Q(2d)$ .

Proof. For $\lambda =(\lambda _1,\cdots ,\lambda _r)\neq (d)$ , we have $r\geq 2$ . By expanding $a_{\lambda }$ and $b_{\lambda }$ except for the first column, we can write $Q_{\lambda }$ as a linear sum of

where $\sigma \in \mathfrak {S}_{2d-r}$ . The latter diagram is obtained from Q by composing a morphism of $\Bbbk \mathbf {F}^{\mathrm {op}}(2d,2d)$ , so is included in $A_d Q(2d)$ .

By Lemma 8.4, we have $A_d Q(n)\supset \sum _{\lambda \vdash d, \lambda \neq (d)}A_d Q_{\lambda }(n)$ . Moreover, we have the following corollary.

Corollary 8.5. The $\operatorname {Aut}(F_n)$ -module $A_d Q(n)$ is generated by $\{Q_{\lambda }\mid \lambda \vdash d, \lambda \neq (d)\}$ for $n\geq 2d$ . That is, we have $A_d Q(n)=\sum _{\lambda \vdash d, \lambda \neq (d)}A_d Q_{\lambda }(n)$ .

Proof. For simplicity, let A denote $\sum _{\lambda \vdash d, \lambda \neq (d)}A_d Q_{\lambda }(n)$ . By Lemma 8.3, we have $\theta _{d,n}(A_d P(n))= B_{d,0}(n)_{(2d)}$ . Thus, by Theorem 8.2, we have

$$ \begin{align*}\theta_{d,n}(A_d Q(n))= \left(\bigoplus_{\lambda\vdash d, \lambda\neq (d)}B_{d,0}(n)_{2\lambda}\right)\oplus \left(\bigoplus_{k\geq 1}B_{d,k}(n)\right).\end{align*} $$

On the other hand, by the PBW theorem, we have

$$ \begin{align*}\theta_{d,n}(A)\supset \left(\bigoplus_{\lambda\vdash d, \lambda\neq (d)}B_{d,0}(n)_{2\lambda}\right).\end{align*} $$

By Corollary 7.9 and Proposition 7.8, we have

$$ \begin{align*}\theta_{d,n}(A)\supset \left(\bigoplus_{\lambda\vdash d, \lambda\neq (d)}B_{d,0}(n)_{2\lambda}\right)\oplus \left(\bigoplus_{k\geq 1}B_{d,k}(n)\right).\end{align*} $$

Therefore, we have $A_d Q(n)\subset A.$ Hence, we have $A_d Q(n)= A$ .

8.2 Radical filtration of $A_d(n)$

For an $\operatorname {Aut}(F_n)$ -module M, let $\operatorname {Rad}(M)$ denote the radical of M; that is,

$$ \begin{gather*} \begin{aligned} \operatorname{Rad}(M) =\bigcap\{K\subset M\mid K \text{ is maximal in }M\}. \end{aligned} \end{gather*} $$

We have a radical filtration of $A_d(n)$

$$ \begin{align*}A_d(n)\supset \operatorname{Rad}(A_d(n))\supset \operatorname{Rad}^2(A_d(n))=\operatorname{Rad}(\operatorname{Rad}(A_d(n)))\supset\cdots.\end{align*} $$

Theorem 8.6. Let $n\geq 2d$ . Then, the filtration of $A_d(n)$ by the number of trivalent vertices coincides with the radical filtration. That is, we have $\operatorname {Rad}(A_{d,k}(n))=A_{d,k+1}(n)$ for any $k\geq 0$ .

Proof. For $\lambda \vdash 2d-k$ , we have $B_{d,k}(n)_{\lambda }\cong \bigoplus _{i=1}^{r_{\lambda }}(V_{\lambda })_{i}$ as $\operatorname {GL}(n;\mathbb {Z})$ -modules. Let $B_{d,k}(n)_{\lambda , i}\subset B_{d,k}(n)_{\lambda }$ be a $\operatorname {GL}(n;\mathbb {Z})$ -submodule corresponding to $(V_{\lambda })_{i}$ . Let $A_{d,k}(n)_{\lambda , i}\subset A_{d,k}(n)$ be the $\operatorname {Aut}(F_n)$ -submodule generated by $\theta _{d,n}^{-1}(B_{d,k}(n)_{\lambda , i})$ . For each $\lambda \vdash 2d-k, i\in [r_{\lambda }]$ , we have a maximal submodule

$$ \begin{align*}R_{\lambda, i}= \left(\sum_{(\mu,j)\neq (\lambda,i)} A_{d,k}(n)_{\mu, j}\right)+ A_{d,k+1}(n).\end{align*} $$

Since we have $\bigcap _{(\lambda ,i)} R_{\lambda , i}=A_{d,k+1}(n)$ , it follows that $\operatorname {Rad}(A_{d,k}(n))\subset A_{d,k+1}(n)$ .

For any maximal submodule K of $A_{d,k}(n)$ , the quotient $A_{d,k}(n)/K$ is an irreducible $\operatorname {Aut}(F_n)$ -module, which factors through an irreducible polynomial $\operatorname {GL}(n;\mathbb {Z})$ -module. It follows that $\theta _{d,n}(A_{d,k}(n))/\theta _{d,n}(K)$ is isomorphic to one of the irreducible components of the $\operatorname {GL}(n;\mathbb {Z})$ -module $\bigoplus _{i\geq k}B_{d,i}(n)$ . If $B_{d,k}(n)\subset \theta _{d,n}(K)$ , then by Proposition 7.8, we have $K= A_{d,k}(n)$ , which contradicts to the maximality of K. Therefore, $\theta _{d,n}(A_{d,k}(n))/\theta _{d,n}(K)$ is isomorphic to one of the irreducible components of $B_{d,k}(n)$ , and we have $K\supset A_{d,k+1}(n)$ . This implies that $\operatorname {Rad}(A_{d,k}(n))\supset A_{d,k+1}(n)$ , and the proof is complete.

It is possible that Theorem 8.6 holds for some $n<2d$ . However, it does not hold for all n. (See Remark 8.13.)

8.3 Indecomposability of the decomposition of $A_d(n)$

Here, we consider the indecomposability of the decomposition (8.1) of $A_d(n)$ .

In Proposition 7.7, we observed that

$$ \begin{align*}B_{d,0}(n)\cong \bigoplus_{\lambda\vdash d}B_{d,0}(n)_{2\lambda},\quad B_{d,1}(n)\cong \bigoplus_{\mu\vdash 2d-1 \text{ with exactly }3\text{ odd parts}}B_{d,1}(n)_{\mu}.\end{align*} $$

In order to study the indecomposability of equation (8.1), we observe certain connectivity at the level of partitions.

Let $X_d=\{ 2\lambda \mid \lambda \vdash d, \lambda \neq (d)\}$ and $Y_d=\{\mu \vdash 2d-1\mid \mu \text { has exactly }3\text { odd parts}\}$ . We consider the bipartite graph $G_d$ with vertex sets $X_d$ and $Y_d$ and with an edge between each pair of vertices $2\lambda $ and $\mu $ if $\mu $ is obtained from $2\lambda $ by taking away one box from each of two different rows of $2\lambda $ and then by adding one box to another row. For example, $G_2$ is

$G_3$ is

and $G_4$ is

Proposition 8.7. The graph $G_d$ is path-connected.

Proof. For $\lambda \vdash d, \lambda \neq (d)$ , let $r(\lambda )$ be the number of rows of $\lambda $ . We write $\lambda =(\lambda _1^{a_1},\lambda _2^{a_2},\cdots ,\lambda _l^{a_l})$ , where $\lambda _1>\lambda _2>\cdots >\lambda _l$ , $\sum _{i=1}^l a_i=r(\lambda )$ , $a_i\geq 1$ .

We show that for $\lambda \vdash d$ such that $r(\lambda )<d$ , there is a path between $2\lambda $ and some $2\lambda '\in X_d$ such that $r(\lambda ')=r(\lambda )+1$ . Then, since $(2^d)$ is the only partition that has d rows, it follows by induction on $k=r(\lambda )$ that all vertices in $X_d$ are path-connected.

If $a_1=k$ , then we have $2\lambda =((2\lambda _1)^k)$ and $2\lambda _1\geq 4$ because we assume that $k<d$ . Thus, we have

$$ \begin{align*}2\lambda {{-}\kern-.2em{-}\kern-.2em{-}\kern-.2em{-}\kern-.2em{-}} \mu', \end{align*} $$

where $\mu '=((2\lambda _1)^{k-2},(2\lambda _1-1)^2,1)$ is obtained from $2\lambda $ by taking away a box from each of the $(k-1)$ -st and k-th row and adding one box to the $(k+1)$ -st row, and

$$ \begin{align*}\mu' {{-}\kern-.2em{-}\kern-.2em{-}\kern-.2em{-}\kern-.2em{-}} 2\lambda', \end{align*} $$

where $2\lambda '=((2\lambda _1)^{k-1},2\lambda _1-2,2)$ is obtained from $\mu '$ by taking away a box from the k-th row and adding a box to each of the $(k-1)$ -st and $(k+1)$ -st row. Therefore, we have a path between $2\lambda $ and $2\lambda '$ such that $r(\lambda ')=k+1$ .

If $a_1< k$ , then we have

$$ \begin{align*}2\lambda {{-}\kern-.2em{-}\kern-.2em{-}\kern-.2em{-}\kern-.2em{-}} \mu'', \end{align*} $$

where $\mu ''$ is obtained from $2\lambda $ by taking away a box from each of the $a_1$ -th and $(a_1+a_2)$ -th row, and adding a box to the $(k+1)$ -st row, and

$$ \begin{align*}\mu''{{-}\kern-.2em{-}\kern-.2em{-}\kern-.2em{-}\kern-.2em{-}} 2\lambda'', \end{align*} $$

where $2\lambda ''$ is obtained from $\mu ''$ by taking away a box from the $a_1$ -th row and adding a box to each of the $(a_1+a_2)$ -th and $(k+1)$ -st row. Therefore, we have a path between $2\lambda $ and $2\lambda ''$ such that $r(\lambda '')=k+1$ .

Lastly, we will show that each vertex of $Y_d$ is connected to a vertex of $X_d$ . Any element $\mu \in Y_d$ is a partition of $2d-1$ and has three odd parts. Therefore, by taking away a box from the last odd row and then adding one box to each of the other two odd rows, we obtain a partition of $2d$ with only even parts, which is a vertex of $X_d$ . The proof is complete.

If $n\geq d$ , then for any $2\lambda \in X_d$ , $B_{d,0}(n)_{2\lambda }$ is a nonzero $\operatorname {GL}(n;\mathbb {Z})$ -submodule of $B_d(n)$ . If $n\geq d$ , then for any $\mu \in Y_d$ (except $\mu =(2^{d-2},1^3)$ if $n=d$ ), $B_{d,1}(n)_{\mu }$ is a nonzero $\operatorname {GL}(n;\mathbb {Z})$ -submodule of $B_d(n)$ .

Let $\pi _{\mu }:B_{d,1}(n)\twoheadrightarrow B_{d,1}(n)_{\mu }$ be the projection.

Proposition 8.8. Let $n\geq 2d$ . Let $2\lambda \in X_d$ , $\mu \in Y_d$ be two endpoints of an edge of the bipartite graph $G_d$ . Then the composition of the bracket map and the projection $\pi _{\mu }$

(8.3) $$ \begin{align} B_{d,0}(n)_{2\lambda}\otimes \operatorname{gr}^1(\operatorname{IA}(n))\xrightarrow{[\cdot,\cdot]} B_{d,1}(n)\xrightarrow{\pi_{\mu}} B_{d,1}(n)_{\mu} \end{align} $$

does not vanish.

Note that this proposition holds for $d=1,2$ because we have $X_1=Y_1=\emptyset , X_2=\{(2^2)\}, Y_2=\{(1^3)\}$ and by Lemma 6.7 in [16].

Recall that we have

and

What the bracket map does is to contract two of the univalent vertices of a diagram of an element of $B_{d,0}(n)$ with two leaves of a trivalent tree in $\operatorname {gr}^1(\operatorname {IA}(n))$ , which corresponds to the operation on partitions of taking away two boxes from different rows and then adding a box. Here, we introduce an intermediate vector space $B_d'(n)$ between $B_{d,0}(n)$ and $B_{d,1}(n)$ , whose elements correspond to partitions which are obtained by the operation of taking away two boxes from different rows. Define $B_d'(n)$ by

where

is a based trivalent tree of degree $1$ . Then, $B_d'(n)$ is a $\operatorname {GL}(n;\mathbb {Z})$ -module, and we have an irreducible decomposition

$$ \begin{align*}B_d'(n)\cong \mathbb{S}_{(d-2)}(\mathbb{S}_{(2)} V_n)\otimes \mathbb{S}_{(1^2)} V_n \cong \bigoplus_{\nu\vdash 2d-2 \text{ with exactly }2\text{ odd parts}} V_{\nu}\end{align*} $$

in a way similar to Proposition 7.7. Let $B_d'(n)_{\nu }$ be the isotypic component of $B_d'(n)$ corresponding to $\nu $ .

Recall that $a_{\lambda }, b_{\lambda }$ and $\diamond $ are defined in Section 5.1. In the proof of Proposition 8.8, we use the following notation

which represents the linear sum of permutations $a_{2\lambda }$ .

Proof of Proposition 8.8 Let $2\lambda =(2\lambda _1,\cdots ,2\lambda _r)\vdash 2d \in X_d$ . Any vertex $\mu \in Y_d$ that is connected to $2\lambda $ by an edge of $G_d$ is obtained from $2\lambda $ by taking away a box from each of the i-th and j-th row of $2\lambda $ and adding a box to the k-th row of $2\lambda $ for some $i,j\in [r], i<j, k\in [r+1], k\neq i,j$ . We write $\mu =(\mu _1,\cdots ,\mu _s)$ . Then we have $\mu _i=2\lambda _i-1, \mu _j=2\lambda _j-1, \mu _k=2\lambda _k+1$ and $\mu _l=2\lambda _l$ for $l\in [s], l\neq i,j,k$ .

Since we have $\operatorname {gr}^1(\operatorname {IA}(n))\cong H^{\ast }\otimes \mathcal {L}_2(n)$ , we can write equation (8.3) by

$$ \begin{align*}h_{\lambda,\mu}:B_{d,0}(n)_{2\lambda}\otimes H^{\ast}\otimes \mathcal{L}_2(n)\rightarrow B_{d,1}(n) \xrightarrow{\pi_{\mu}} B_{d,1}(n)_{\mu}.\end{align*} $$

We will show that $h_{\lambda ,\mu }$ does not vanish.

Let $\nu \vdash 2d-2$ be the partition that is obtained from $2\lambda $ by taking away a box from each of the i-th and j-th row of $2\lambda $ . We decompose $h_{\lambda ,\mu }$ into the composition

$$ \begin{align*}h_{\lambda,\mu}=h_{\nu,\mu} h_{\lambda,\nu},\end{align*} $$

where $h_{\nu ,\mu }$ and $h_{\lambda ,\nu }$ are $\operatorname {GL}(n;\mathbb {Z})$ -module maps defined as follows.

Let

$$ \begin{align*}h^{\prime}_{\lambda}:B_{d,0}(n)_{2\lambda}\otimes\mathcal{L}_2(n)\rightarrow B_d'(n)\end{align*} $$

be a $\operatorname {GL}(n;\mathbb {Z})$ -module map defined in a way similar to the contraction map in Section 5.2. Define

$$ \begin{align*}h_{\lambda}:B_{d,0}(n)_{2\lambda}\otimes H^{\ast}\otimes \mathcal{L}_2(n)\rightarrow B_d'(n)\otimes H^{\ast}\end{align*} $$

by $h_{\lambda }(x\otimes y\otimes z)=h^{\prime }_{\lambda }(x\otimes z)\otimes y$ for $x\in B_{d,0}(n)_{2\lambda }, y\in H^{\ast }, z\in \mathcal {L}_2(n)$ . We also define a $\operatorname {GL}(n;\mathbb {Z})$ -module map

$$ \begin{align*}h:B_d'(n)\otimes H^{\ast}\rightarrow B_{d,1}(n)\end{align*} $$

by connecting two bases $\ast _1,\ast _2$ , that is, for $w_1,\cdots , w_{2d-2}\in V_n, v\in H^{\ast }$ ,

Let $\pi _{\nu }: B_d'(n)\otimes H^{\ast }\rightarrow B_d'(n)_{\nu }\otimes H^{\ast }$ be the tensor product of the projection and $\operatorname {id}_{H^{\ast }}$ . Then we have two $\operatorname {GL}(n;\mathbb {Z})$ -module maps

$$ \begin{align*}h_{\lambda, \nu}:B_{d,0}(n)_{2\lambda}\otimes H^{\ast}\otimes \mathcal{L}_2(n)\xrightarrow{h_{\lambda}} B_d'(n)\otimes H^{\ast}\xrightarrow{\pi_{\nu}} B_d'(n)_{\nu}\otimes H^{\ast}\end{align*} $$

and

$$ \begin{align*}h_{\nu,\mu}:B_d'(n)_{\nu}\otimes H^{\ast}\xrightarrow{h} B_{d,1}(n)\xrightarrow{\pi_\mu} B_{d,1}(n)_{\mu}.\end{align*} $$

Since $h_{\lambda ,\nu }$ and $h_{\nu ,\mu }$ are $\operatorname {GL}(n;\mathbb {Z})$ -module maps and since $B_{d,0}(n)_{2\lambda }$ and $B_d'(n)_{\nu }$ are irreducible, it suffices to prove that $h_{\lambda ,\nu }\neq 0$ and $h_{\nu ,\mu }\neq 0$ .

We will prove that $h_{\lambda ,\nu }$ does not vanish. Let

where $\bar {i}=\sum _{l=1}^{i}2\lambda _l-1, \bar {j}=\sum _{l=1}^{j}2\lambda _l-2$ . Since we have

$$ \begin{align*}c_{\nu}\diamond c_{(1^2)}\in S^{\nu}\diamond S^{(1^2)}=\bigoplus_{\rho\vdash 2d}(S^{\rho})^{LR^{\rho}_{\nu,(1^2)}}\end{align*} $$

and

$$ \begin{align*}\{\rho\vdash 2d \mid LR^{\rho}_{\nu,(1^2)}\neq 0 \}\cap X_d=\{2\lambda\},\end{align*} $$

we have $u\in B_{d,0}(n)_{2\lambda }$ . Moreover, we have

(8.4)

By the relation $b_{(1^2)}=\operatorname {id}-(1,2)$ and the AS relation, the right-hand side of equation (8.4) is

Since we have

locally, by pulling $\ast _1$ to the top, we have

We will look at the coefficient in $u'$ of

to show that $u'$ does not vanish. Note that the upper box corresponds to $a_{\nu }$ and that $b_{\nu }a_{\nu }=\sum _{\tau \in C_{t_0},\rho \in R_{t_0}} \mathrm {sgn}(\tau )\tau \rho $ , where $t_0$ is the canonical $\nu $ -tableau. If $\lambda _i\neq \lambda _j$ , then there is no $\tau \in C_{t_0}$ such that $\tau (\bar {i})=\bar {j}, \tau (\bar {j})=\bar {i}$ . Thus, the diagram $u_0$ appears only when $\tau $ is an even permutation which fixes $\bar {i}$ and $\bar {j}$ . Then, the coefficient of $u_0$ in $u'$ is negative. If $\lambda _i= \lambda _j$ , then the diagram $u_0$ appears when $\tau $ preserves the subset $\{\bar {i},\bar {j}\}$ and the parity of $\tau $ coincides with that of the restriction of $\tau $ to $\{\bar {i},\bar {j}\}$ . Hence, by the AS relation, the coefficient of $u_0$ in $u'$ is negative. Therefore, $h_{\lambda , \nu }$ does not vanish.

We will prove that $h_{\nu ,\mu }$ does not vanish. Let $N\in \mathbb {N}$ . Set $c^{\prime }_{\rho }=a_{\rho } b_{\rho }\in \Bbbk \mathfrak {S}_{N}$ for $\rho \vdash N$ . From basic facts of representation theory, we have an isomorphism of $\Bbbk \mathfrak {S}_{N}$ -modules

$$ \begin{align*}\Bbbk\mathfrak{S}_{N}c_{\rho} \cong \Bbbk\mathfrak{S}_{N}c^{\prime}_{\rho}.\end{align*} $$

In what follows, we use $c^{\prime }_{\rho }$ instead of $c_{\rho }$ as the Young symmetrizer. Let

where $\sigma \in \mathfrak {S}_{2d-1}$ is defined by

$$ \begin{align*}\sigma= \begin{pmatrix} 1&\cdots &2d-3 &2d-2 & 2d-1\\ 1&\cdots &i' &j' & k' \end{pmatrix} \text{ for } i'=\sum_{l=1}^{i}\mu_{l},\: j'=\sum_{l=1}^{j} \mu_{l},\: k'=\sum_{l=1}^{k} \mu_{l}. \end{align*} $$

We will show that $h(\pi _{\nu }(Z_{\mu }))\in B_{d,1}(n)_{\mu }$ and that $h(\pi _{\nu }(Z_{\mu }))\neq 0$ .

If the diagram that is obtained from $\mu $ by taking away a box from the i-th (resp. j-th) row of $\mu $ is a partition of $2d-2$ , then write it $\nu _i$ (resp. $\nu _j$ ). Since any partition $\rho \vdash 2d-2$ with exactly two odd parts other than $\nu ,\nu _i,\nu _j$ is not included in $\mu $ , it follows that

$$ \begin{align*}Z_\mu \in (B_d'(n)_{\nu}\otimes H^{\ast}) \oplus (B_d'(n)_{\nu_i}\otimes H^{\ast}) \oplus(B_d'(n)_{\nu_j}\otimes H^{\ast}).\end{align*} $$

By using an argument similar to Proposition 5.3, we have

$$ \begin{gather*} h(B_d'(n)_{\nu}\otimes H^{\ast})\subset \bigoplus_{\alpha=\nu\sqcup \square} B_{d,1}(n)_{\alpha},\quad h(B_d'(n)_{\nu_i}\otimes H^{\ast})\subset \bigoplus_{\alpha=\nu_i\sqcup \square} B_{d,1}(n)_{\alpha},\\ h(B_d'(n)_{\nu_j}\otimes H^{\ast})\subset \bigoplus_{\alpha=\nu_j\sqcup \square} B_{d,1}(n)_{\alpha}. \end{gather*} $$

Since $\{\nu \sqcup \square \}\cap \{\nu _i\sqcup \square \}\cap \{\nu _j\sqcup \square \}=\{\mu \}$ and since $h(Z_{\mu })\in B_{d,1}(n)_{\mu }$ , we have $h(\pi _{\nu }(Z_{\mu }))\in B_{d,1}(n)_{\mu }$ .

In order to prove that $h(\pi _{\nu }(Z_{\mu }))\neq 0$ , we will look at the coefficient in $h(\pi _{\nu }(Z_{\mu }))$ of

Note that $c^{\prime }_{\mu }=\sum _{\rho \in R_{s_0},\tau \in C_{s_0}} \mathrm {sgn}(\tau )\rho \tau $ , where $s_0$ is the canonical $\mu $ -tableau.

Firstly, we consider the case where $\mu _i,\mu _j,\mu _k$ are distinct. Then z appears only when $\tau $ is an even permutation which fixes $i'$ , $j'$ and $k'$ . Therefore, the coefficient of z in $h(Z_{\mu })$ is positive. Moreover, the linear sum of terms in $Z_{\mu }$ such that $\ast _2$ is connected to $v_k$ lies in $\pi _{\nu }(Z_{\mu })$ , so the coefficient of z in $h(\pi _{\nu }(Z_{\mu }))$ is equal to that of z in $h(Z_{\mu })$ , which is nonzero.

The other cases, where at least two of $\mu _i,\mu _j$ and $\mu _k$ are equal, follow in a similar argument. The only thing that differs from the above case is that z appears when $\tau $ preserves the subset $\{i',j',k'\}\subset [2d-1]$ , and the parity of $\tau $ coincides with that of the restriction of $\tau $ to $\{i',j',k'\}$ . Since we have the AS relation, the sign due to the permutation of $\{i',j',k'\}$ is cancelled. Therefore, the coefficient of z in $h(Z_{\mu })$ is positive in any case. The proof is complete.

Theorem 8.9. Let $d\geq 2$ . The direct decomposition

$$ \begin{align*}A_d(n)=A_d P(n)\oplus A_d Q(n)\end{align*} $$

of $\operatorname {Aut}(F_n)$ -modules is indecomposable for $n\geq 2d$ .

Proof. By Lemma 8.3, it suffices to show that $A_d Q(n)$ is indecomposable. Since the radical preserves the direct sum, we have only to show that $A_d Q(n)/\operatorname {Rad}^2(A_d Q(n))$ is indecomposable. Suppose that we have a nontrivial decomposition of $\operatorname {Aut}(F_n)$ -modules

$$ \begin{gather*} \begin{aligned} A_d Q(n)/\operatorname{Rad}^2(A_d Q(n))&=A_d Q(n)/A_{d,2}(n)\\ &=(M_1+A_{d,2}(n))/ A_{d,2}(n)\oplus (M_2+A_{d,2}(n))/A_{d,2}(n), \end{aligned} \end{gather*} $$

where $M_i$ is an $\operatorname {Aut}(F_n)$ -submodule of $A_d Q(n)$ for $i=1,2$ . Let

$$ \begin{align*}N_i=\theta_{d,n}(M_i+A_{d,2}(n))/\theta_{d,n}(A_{d,2}(n))\end{align*} $$

for $i=1,2$ . We have

$$ \begin{align*}N_1\oplus N_2=\theta_{d,n}(A_d Q(n))/\theta_{d,n}(A_{d,2}(n))=\left(\bigoplus_{\lambda\vdash d, \lambda\neq (d)} B_{d,0}(n)_{2\lambda}\right) \oplus B_{d,1}(n).\end{align*} $$

For any $2\lambda \in X_d$ , there uniquely exists $i\in \{1,2\}$ such that $N_i$ includes a $\operatorname {GL}(n;\mathbb {Z})$ -submodule $(N_i)_{2\lambda }\cong V_{2\lambda }$ . Let $x\in (N_i)_{2\lambda }$ be a generator of the irreducible $\operatorname {GL}(n;\mathbb {Z})$ -module $(N_i)_{2\lambda }$ . Then, the image $x'$ of x under the composition of $\operatorname {GL}(n;\mathbb {Z})$ -module maps

$$ \begin{align*}(N_i)_{2\lambda}\hookrightarrow N_i \hookrightarrow B_{d,0}(n)\oplus B_{d,1}(n)\twoheadrightarrow B_{d,0}(n)\end{align*} $$

is an element of $B_{d,0}(n)_{2\lambda }$ . For any $\mu \in Y_d$ that is connected to $2\lambda $ by an edge of $G_d$ , by Proposition 8.8, there exists $g\in \operatorname {gr}^1(\operatorname {IA}(n))$ such that $[x',g]\neq 0\in B_{d,1}(n)_{\mu }$ . Therefore, we have

$$ \begin{align*}[x,g]=[x',g]+[x-x',g]=[x',g]\neq 0\in B_{d,1}(n)_{\mu}.\end{align*} $$

It follows that $N_i$ includes a $\operatorname {GL}(n;\mathbb {Z})$ -submodule $(N_i)_{\mu }$ that is isomorphic to $V_{\mu }$ for any $\mu \in Y_d$ that is connected to $2\lambda $ by an edge of $G_d$ . Hence, by Proposition 8.7, we have $N_1\cap N_2\neq \{0\}$ , a contradiction. Therefore, $A_d Q(n)$ is indecomposable.

Note that the assumption $n\geq 2d$ is needed for the surjectivity of the bracket map and the nontriviality of the bracket map for each pair of nonzero irreducible $\operatorname {GL}(n;\mathbb {Z})$ -submodules. Thus, if we have the surjectivity and the nontriviality of the bracket map for some $n<2d$ , we can loose the assumption.

8.4 The $\operatorname {Aut}(F_n)$ -module structure of $A_3(n)$

Here, we consider the $\operatorname {Aut}(F_n)$ -module structure of $A_3(n)$ in detail.

In degree $3$ , the restrictions of the bracket map to each isotypic component induce $\operatorname {GL}(n;\mathbb {Z})$ -module homomorphisms

$$ \begin{gather*} \begin{aligned} \rho_1:B_{3,0}(n)_{(4,2)}&\rightarrow \operatorname{Hom}(\operatorname{gr}^1(\operatorname{IA}(n)), B_{3,1}(n)_{(3,1^2)}),\\ \rho_2:B_{3,0}(n)_{(2^3)}&\rightarrow \operatorname{Hom}(\operatorname{gr}^1(\operatorname{IA}(n)), B_{3,1}(n)_{(3,1^2)}),\\ \rho_3:B_{3,0}(n)_{(2^3)}&\rightarrow \operatorname{Hom}(\operatorname{gr}^1(\operatorname{IA}(n)), B_{3,1}(n)_{(2,1^3)}). \end{aligned} \end{gather*} $$

Proposition 8.10. The $\operatorname {GL}(n;\mathbb {Z})$ -module homomorphisms $\rho _1$ and $\rho _2$ are injective for $n\geq 3$ and $\rho _3$ for $n\geq 4$ .

Proof. Recall that $c_{\lambda }$ denotes the Young symmetrizer defined in equation (5.1) and that $K_{i,j,k}\in \operatorname {IA}(n)$ is defined by equation (7.4). For $n\geq 3$ , we have

$$ \begin{align*}\rho_1(u)(K_{3,2,1})=[u,K_{3,2,1}]=-10 w\neq 0 \in B_{3,1}(n)_{(3,1^2)},\end{align*} $$

where

and

Thus, we have $\rho _1\neq 0$ for $n\geq 3$ . Since $B_{3,0}(n)_{(4,2)}$ is irreducible, $\rho _1$ is injective.

Let

We have

$$ \begin{align*}\rho_2(x)(K_{1,3,2})=[x,K_{1,3,2}]= -6 w\neq 0 \in B_{3,1}(n)_{(3,1^2)}.\end{align*} $$

Thus, we have $\rho _2\neq 0$ for $n\geq 3$ . Since $B_{3,0}(n)_{(2^3)}$ is irreducible, $\rho _2$ is injective.

For $n\geq 4$ , we have

$$ \begin{align*}[x,K_{4,3,2}]=-\frac{6}{5}y-\frac{24}{5}z,\end{align*} $$

and thus,

$$ \begin{align*}\rho_3(x)(K_{4,3,2})=-\frac{24}{5}z\neq 0 \in B_{3,1}(n)_{(2,1^3)},\end{align*} $$

where

and

Therefore, we have $\rho _3\neq 0$ for $n\geq 4$ . Since $B_{3,0}(n)_{(2^3)}$ is irreducible, $\rho _3$ is injective.

Remark 8.11. We consider a restriction of the bracket map

(8.5) $$ \begin{align} [\cdot,\cdot]:V_{\lambda}\otimes \operatorname{gr}^1(\operatorname{IA}(n))\rightarrow V_{\mu} \end{align} $$

for each irreducible $\operatorname {GL}(n;\mathbb {Z})$ -submodule $V_{\lambda }$ (resp. $V_{\mu }$ ) of $B_{d,k}(n)$ (resp. $B_{d,k+1}(n)$ ). We write a wavy arrow

$$ \begin{align*}V_{\lambda}\rightsquigarrow V_{\mu}\end{align*} $$

if the restriction map (8.5) does not vanish. Then, we have the following diagram for $n\geq 4$ :

where $B_{3,2}^{(i)}(n)_{(2^2)}$ is the irreducible component of $B_{3,2}(n)_{(2^2)}$ generated by

respectively. Note that, for $n=3$ , $B_3(3)$ includes all of the above irreducible subrepresentations but $B_{3,1}(3)_{(2,1^3)}=0$ , and there are all of the wavy arrows but the three wavy arrows that are directed to or coming from $B_{3,1}(3)_{(2,1^3)}$ . For $n=2$ , we have

$$ \begin{align*}B_3(2)=(B_{3,0}(2)_{(6)} \oplus B_{3,0}(2)_{(4,2)})\oplus B_{3,2}(2)\oplus B_{3,4}(2).\end{align*} $$

For $n=1$ , we have

$$ \begin{align*}B_3(1)=B_{3,0}(1)_{(6)} \oplus B_{3,2}(1)_{(4)} \oplus B_{3,4}(1)_{(2)}.\end{align*} $$

For $n=1,2$ , there are no wavy arrows because $B_{3,1}(n)=B_{3,3}(n)=0$ .

By Proposition 8.10 and Remark 8.11, we have the surjectivity and the nontriviality of the bracket map for $n\geq 3$ . Thus, by Theorem 8.9, one can obtain the following theorem, which improves Theorem 8.9 for $d=3$ .

Theorem 8.12. We have an indecomposable decomposition

$$ \begin{align*}A_3(n)=A_3 P(n)\oplus A_3 Q(n)\end{align*} $$

of $\operatorname {Aut}(F_n)$ -modules for $n\geq 3$ .

For $\lambda \vdash 4$ , let , where $c_{\lambda }$ is the Young symmetrizer. Let and .

For $n=2$ , we can check that $A_{3,2}(2)$ is semisimple as $\operatorname {Aut}(F_2)$ -modules, that is,

$$ \begin{align*}A_{3,2}(2)=A_3 R_{(4)}(2)\oplus A_3 R_{(3,1)}(2)\oplus A_3 S(2)\oplus A_3 U(2)\oplus A_3 T(2),\end{align*} $$

where We do not know whether or not the $\operatorname {Aut}(F_2)$ -module $A_3(2)$ is semisimple.

Remark 8.13. Since $A_{3,2}(2)$ is semisimple, we have $\operatorname {Rad}(A_{3,2}(2))=0$ . On the other hand, we have $A_{3,3}(2)=A_{3,4}(2)\cong B_{3,4}(2)\neq 0$ . Therefore, we have $\operatorname {Rad}(A_{3,2}(2))\neq A_{3,3}(2)$ .

For $n=1$ , we have $\operatorname {Aut}(F_1)=\mathbb {Z}/2\mathbb {Z}$ . We can easily check the following proposition.

Proposition 8.14. The $\operatorname {Aut}(F_1)$ -action on $A_3(1)$ is trivial. Therefore, we have $A_3(1)=A_3 P(1)\oplus A_3 R_{(4)}(1) \oplus A_3 T(1).$

8.5 The socle of $A_d(n)$ for small d

For an $\operatorname {Aut}(F_n)$ -module M, let $\operatorname {Soc}(M)$ denote the socle of M; that is,

$$ \begin{align*}\operatorname{Soc}(M)=\sum\{K\subset M \mid K \text{ is simple}\}.\end{align*} $$

Let us consider the cases for small d. Since $A_1(n)\cong \operatorname {Sym}^2(V_n)$ is simple, we have

$$ \begin{align*}\operatorname{Soc}(A_1(n))=A_1(n)\quad (n\geq 1).\end{align*} $$

By Theorem 6.9 of [16], we have

$$ \begin{align*}\operatorname{Soc}(A_2(n))=A_2 P(n)\oplus A_2 \tilde{T}(n) \quad(n\geq 3, n=1),\end{align*} $$

$$ \begin{align*}\operatorname{Soc}(A_2(n))=A_2(n)=A_2 P(n)\oplus A_2 W(n) \oplus A_2 \tilde{T}(n) \quad(n=2),\end{align*} $$

where

Note that $A_2 \tilde {T}(n)=A_{2,2}(n)$

By Proposition 8.14, we have $\operatorname {Soc}(A_3(1))=A_3(1)$ .

Proposition 8.15. For $n\geq 3$ , we have

$$ \begin{align*}\operatorname{Soc}(A_3(n))=A_3 P(n)\oplus A_3 R_{(4)}(n)\oplus A_3 R_{(3,1)}(n)\oplus A_3 S(n)\oplus A_3 T(n).\end{align*} $$

Proof. A simple $\operatorname {Aut}(F_n)$ -submodule $K\subset A_3(n)$ corresponds to an irreducible component of $B_3(n)$ via the PBW map. Therefore, by Remark 8.11, we have

$$ \begin{align*}\operatorname{Soc}(A_3(n))\subset A_3 P(n)\oplus A_3 R_{(4)}(n)\oplus A_3 R_{(3,1)}(n)\oplus A_3 S(n)\oplus A_3 T(n).\end{align*} $$

Moreover, we can check that

$$ \begin{gather*} A_3 P(n)\cong V_{(6)},\quad A_3 R_{(4)}(n)\cong V_{(4)},\quad A_3 R_{(3,1)}(n)\cong V_{(3,1)},\\ A_3 S(n)\cong V_{(2,2)},\quad A_3 T(n)\cong V_{(2)}. \end{gather*} $$

Hence, we have

$$ \begin{align*}\operatorname{Soc}(A_3(n))\supset A_3 P(n)\oplus A_3 R_{(4)}(n)\oplus A_3 R_{(3,1)}(n)\oplus A_3 S(n)\oplus A_3 T(n),\end{align*} $$

and the proof is complete.

8.6 The indecomposable decomposition of $A_4(n)$

Here, we consider the indecomposable decomposition of $A_4(n)$ .

Similarly, in degree $4$ , we have $\operatorname {GL}(n;\mathbb {Z})$ -module homomorphisms

$$ \begin{gather*} \begin{aligned} \rho_1:B_{4,0}(n)_{(6,2)}&\rightarrow \operatorname{Hom}(\operatorname{gr}^1(\operatorname{IA}(n)), B_{4,1}(n)_{(5,1^2)}),\\ \rho_2:B_{4,0}(n)_{(4^2)}&\rightarrow \operatorname{Hom}(\operatorname{gr}^1(\operatorname{IA}(n)), B_{4,1}(n)_{(3^2,1)}),\\ \rho_3:B_{4,0}(n)_{(4,2^2)}&\rightarrow \operatorname{Hom}(\operatorname{gr}^1(\operatorname{IA}(n)), B_{4,1}(n)_{(5,1^2)}),\\ \rho_4:B_{4,0}(n)_{(4,2^2)}&\rightarrow \operatorname{Hom}(\operatorname{gr}^1(\operatorname{IA}(n)), B_{4,1}(n)_{(4,1^3)}),\\ \rho_5:B_{4,0}(n)_{(4,2^2)}&\rightarrow \operatorname{Hom}(\operatorname{gr}^1(\operatorname{IA}(n)), B_{4,1}(n)_{(3^2,1)}),\\ \rho_6:B_{4,0}(n)_{(4,2^2)}&\rightarrow \operatorname{Hom}(\operatorname{gr}^1(\operatorname{IA}(n)), B_{4,1}(n)_{(3,2,1^2)}),\\ \rho_7:B_{4,0}(n)_{(2^4)}&\rightarrow \operatorname{Hom}(\operatorname{gr}^1(\operatorname{IA}(n)), B_{4,1}(n)_{(3,2,1^2)}),\\ \rho_8:B_{4,0}(n)_{(2^4)}&\rightarrow \operatorname{Hom}(\operatorname{gr}^1(\operatorname{IA}(n)), B_{4,1}(n)_{(2^2,1^3)}). \end{aligned} \end{gather*} $$

Proposition 8.16. The $\operatorname {GL}(n;\mathbb {Z})$ -module homomorphisms $\rho _1, \rho _2,\rho _3$ and $\rho _5$ are injective for $n\geq 3$ , $\rho _4,\rho _6$ and $\rho _7$ for $n\geq 4$ and $\rho _8$ for $n\geq 5$ .

Proof. As in the proof of Proposition 8.10 in degree $3$ , we will check that $\rho _1$ is injective for $n\geq 3$ , $\rho _7$ for $n\geq 4$ and $\rho _8$ for $n\geq 5$ . The others can be obtained in a similar way.

For $n\geq 3$ , we have

$$ \begin{align*}[u,K_{3,1,2}]=14 w\neq 0\in B_{4,1}(n)_{(5,1^2)},\end{align*} $$

where

and

Thus, we have $\rho _1\neq 0$ for $n\geq 3$ . Since $B_{4,0}(n)_{(6,2)}$ is irreducible, $\rho _1$ is injective.

For $n\geq 4$ , we have

$$ \begin{align*}[x,K_{1,4,3}]=-48 y\neq 0 \in B_{4,1}(n)_{(3,2,1^2)},\end{align*} $$

where

and

Thus, $\rho _7$ is injective for $n\geq 4$ .

For $n\geq 5$ , we have

$$ \begin{align*}[x, K_{5,4,3}]=-48y' -32z,\end{align*} $$

where

and

Therefore, we have

$$ \begin{align*}\rho_8(x)(K_{5,4,3})=-32z\neq 0\in B_{4,1}(n)_{(2^2,1^3)},\end{align*} $$

and thus, $\rho _8$ is injective for $n\geq 5$ .

By using Theorem 8.9 and Proposition 8.16 carefully, one can obtain the following theorem, which improves Theorem 8.9 for $d=4$ .

Theorem 8.17. We have an indecomposable decomposition

$$ \begin{align*}A_4(n)=A_4 P(n)\oplus A_4 Q(n)\end{align*} $$

of $\operatorname {Aut}(F_n)$ -modules for $n\geq 7$ .

We expect that Theorem 8.17 holds for $n\geq 3$ .

9 The $\operatorname {Out}(F_n)$ -module structure of $A_d(n)$

In [16], we observed that the $\operatorname {Aut}(F_n)$ -action on $A_d(n)$ induces an action of $\operatorname {Out}(F_n)$ on $A_d(n)$ . In this section, we obtain some results for $A_d(n)$ as $\operatorname {Out}(F_n)$ -modules, which is induced by the results in Section 8.

Since the $\operatorname {Aut}(F_n)$ -action on $A_d(n)$ factors through $\operatorname {Out}(F_n)$ , any submodule of $A_d(n)$ as $\operatorname {Aut}(F_n)$ -modules is a submodule of $A_d(n)$ as $\operatorname {Out}(F_n)$ -modules, and vice versa. By Theorem 8.6, we obtain the radical filtration of $A_d(n)$ as $\operatorname {Out}(F_n)$ -modules.

Theorem 9.1. Let $n\geq 2d$ . Then, the filtration of $A_d(n)$ by the number of trivalent vertices coincides with the radical filtration of $A_d(n)$ as $\operatorname {Out}(F_n)$ -modules.

By Theorem 8.9, we obtain an indecomposable decomposition of $A_d(n)$ as $\operatorname {Out}(F_n)$ -modules.

Theorem 9.2. Let $d\geq 2$ . We have a direct decomposition

$$ \begin{align*}A_d(n)=A_d P(n)\oplus A_d Q(n)\end{align*} $$

of $\operatorname {Out}(F_n)$ -modules, which is indecomposable for $n\geq 2d$ .

Theorems 8.12, 8.17 also hold as $\operatorname {Out}(F_n)$ -modules. Other results for $A_d(n)$ as $\operatorname {Aut}(F_n)$ -modules such as Proposition 8.15 also hold.

10 Indecomposable decomposition of the functor $A_d$

In this section, we obtain an indecomposable decomposition of the functor $A_d$ by using results in Section 8.

By Theorem 8.2, we obtain the following direct decomposition of the functor $A_d$ .

Theorem 10.1. We have a direct decomposition

$$ \begin{align*}A_d=A_d P\oplus A_d Q\end{align*} $$

in the functor category $\mathbf {fVect}^{\mathbf {F}^{\mathrm {op}}}$ .

For $d=1$ , we have $A_1 Q=0$ and the functor $A_1=A_1 P$ is simple. For $d=2$ , we obtained this direct decomposition in Theorem 6.5 of [16]. Moreover, we proved that this direct decomposition of the functor $A_2$ is indecomposable (see Theorem 6.14 of [16]).

By Theorem 8.9, we obtain the indecomposability of the direct decomposition of the functor $A_d$ .

Proposition 10.2. Let $d\geq 2$ . The decomposition

$$ \begin{align*}A_d=A_d P\oplus A_d Q\end{align*} $$

of the functor $A_d$ is indecomposable in the functor category $\mathbf {fVect}^{\mathbf {F}^{\mathrm {op}}}$ .

Proof. Suppose that we have a decomposition

$$ \begin{align*}A_d Q=G\oplus G'\in \mathbf{fVect}^{\mathbf{F}^{\mathrm{op}}}.\end{align*} $$

Then we have $A_d Q(2d)=G(2d)\oplus G'(2d)$ as $\operatorname {Aut}(F_{2d})$ -modules. By Theorem 8.9, the $\operatorname {Aut}(F_{2d})$ -module $A_d Q(2d)$ is indecomposable. Therefore, we can assume that $G'(2d)=0$ and $A_d Q(2d)=G(2d)$ . Since the subfunctor $A_d Q$ is generated by $Q\in A_d Q(2d)$ , we have $A_d Q=G$ . Hence, the subfunctor $A_d Q$ is also indecomposable. By Lemma 8.3, $A_d P(2d)$ is also indecomposable. Therefore, by the similar argument, the subfunctor $A_d P$ is also indecomposable.