We apply Takesaki’s and Connes’s ideas on structure analysis for type III factors to the study of links (a short term of Markov kernels) appearing in asymptotic representation theory.

We categorify the commutation of Nakajima’s Heisenberg operators $P_{\pm 1}$ andtheir infinitely many counterparts in the quantum toroidal algebra $U_{q_1,q_2}(\ddot {gl_1})$ acting on the Grothendieck groups of Hilbert schemes from [10, 24, 26, 32]. By combining our result with [26], one obtains a geometric categorical $U_{q_1,q_2}(\ddot {gl_1})$ action on the derived category of Hilbert schemes. Our main technical tool is a detailed geometric study of certain nested Hilbert schemes of triples and quadruples, through the lens of the minimal model program, by showing that these nested Hilbert schemes are either canonical or semidivisorial log terminal singularities.

Given a free unitary quantum group $G=A_{u}(F)$, with $F$ not a unitary $2\times 2$ matrix, we show that the Martin boundary of the dual of $G$ with respect to any $G$-${\hat{G}}$-invariant, irreducible, finite-range quantum random walk coincides with the topological boundary defined by Vaes and Vander Vennet. This can be thought of as a quantum analogue of the fact that the Martin boundary of a free group coincides with the space of ends of its Cayley tree.

The study of graph ${{C}^{*}}$-algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have not yet been computed. We close this gap by proving that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph ${{C}^{*}}$-algebra. This shows that the quantum symmetry of a graph coincides with the quantum symmetry of the graph ${{C}^{*}}$-algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two definitions) for undirected graphs on four vertices.

We will give a representation-theoretic proof for the multiplication formula in the Ringel-Hall algebra $\mathfrak{H}\vartriangle \,(n)$ of a cyclic quiver $\Delta \,(n)$. As a first application, we see immediately the existence of Hall polynomials for cyclic quivers, a fact established by J. Y. Guo and C. M. Ringel, and derive a recursive formula to compute them. We will further use the formula and the construction of a monomial basis for $\mathfrak{H}\vartriangle \,(n)$ given by Deng, Du, and Xiao together with the double Ringel-Hall algebra realisation of the quantum loop algebra ${{U}_{v}}({{\widehat{\mathfrak{g}\mathfrak{l}}}_{n}})$ given by Deng, Du, and Fu to develop some algorithms and to compute the canonical basis for $U_{v}^{+}({{\widehat{\mathfrak{g}\mathfrak{l}}}_{n}})$. As examples, we will show explicitly the part of the canonical basis associated with modules of Lowey length at most 2 for the quantum group ${{U}_{v}}({{\widehat{\mathfrak{g}\mathfrak{l}}}_{2}})$.

We construct the quantized enveloping algebra of any simple Lie algebra of type $\mathbb{A}\mathbb{D}\mathbb{E}$ as the quotient of a Grothendieck ring arising from certain cyclic quiver varieties. In particular, the dual canonical basis of a one-half quantum group with respect to Lusztig’s bilinear form is contained in the natural basis of the Grothendieck ring up to rescaling. This paper expands the categorification established by Hernandez and Leclerc to the whole quantum groups. It can be viewed as a geometric counterpart of Bridgeland’s recent work for type $\mathbb{A}\mathbb{D}\mathbb{E}$.

We classify the affine actions of ${{U}_{q}}\left( sl\left( 2 \right) \right)$ on commutative polynomial rings in $m\,\ge \,1$ variables. We show that, up to scalar multiplication, there are two possible actions. In addition, for each action, the subring of invariants is a polynomial ring in either $m$ or $m\,-\,1$ variables, depending upon whether $q$ is or is not a root of 1.

We introduce and study a remarkable family of real probability measures ${{\pi }_{st}}$ that we call free Bessel laws. These are related to the free Poisson law $\pi $ via the formulae ${{\text{ }\!\!\pi\!\!\text{ }}_{s1}}={{\text{ }\!\!\pi\!\!\text{ }}^{\boxtimes s}}$ and $\text{ }\pi {{\text{ }}_{1t}}=\text{ }\pi {{\text{ }}^{\boxtimes }}^{t}$. Our study includes definition and basic properties, analytic aspects (supports, atoms, densities), combinatorial aspects (functional transforms, moments, partitions), and a discussion of the relation with random matrices and quantum groups.

In this paper we give a re-normalization of the Reshetikhin–Turaev quantum invariants of links, using modified quantum dimensions. In the case of simple Lie algebras these modified quantum dimensions are proportional to the usual quantum dimensions. More interestingly, we give two examples where the usual quantum dimensions vanish but the modified quantum dimensions are non-zero and lead to non-trivial link invariants. The first of these examples is a class of invariants arising from Lie superalgebras previously defined by the first two authors. These link invariants are multivariable and generalize the multivariable Alexander polynomial. The second example is a hierarchy of link invariants arising from nilpotent representations of quantized at a root of unity. These invariants contain Kashaev’s quantum dilogarithm invariants of knots.

We introduce and study several notions of amenability for unitary corepresentations and $*$-representations of algebraic quantum groups, which may be used to characterize amenability and co-amenability for such quantum groups. As a background for this study, we investigate the associated tensor ${{C}^{*}}$-categories.

Let Γ be one of the N2-dimensional bicovariant first-order differential calculi on the quantum groups Oq(N) or Spq(N), where q is not a root of unity. We show that the second antisymmetrizer exterior algebra sΓ∧ is the quotient of the universal exterior algebra uΓ∧ by the principal ideal generated by θ∧θ. Here ∧ denotes the unique up to scalars bi-invariant 1-form. Moreover, θ∧θ is central in uΓ∧ and uΓ∧ is an inner differential calculus.

Let Λ be a finite-dimensional hereditary algebra over a finite field k, ${\mathcal H}$(Λ) and ${\mathcal C}$(Λ) be, respectively, the Hall algebra and the composition algebra of Λ, ${\mathcal P}$ be the isomorphism classes of finite dimensional Λ-modules and I the isomorphism classes of simple Λ-modules. We define δ$_α$ and $_α$ δ, α in ${\mathcal P}$, to be the right and left derivations of ${\mathcal H}$(Λ) respectively. By using these derivations and the action of the braid group on the set of exceptional sequences of Λ-mod, we provide an effective algorithm of calculating the root vectors of real Schur roots. This means that we get an inductive method to express u$_λ$ as the combinations of elements u$_i$ in the Hall algebra, where i ∈ I and λ in ${\mathcal P}$ is any exceptional Λ-module. Because of the canonical isomorphism between the Drinfeld–Jimbo quantum group and the generic composition algebra, our algorithm is applicable directly to quantum groups. In particular, all the root vectors are obtained in this way in the finite type cases.