Simple diagonal locally finite lie algebras

Abstract

The ground field $F$ is an algebraically closed field of zero characteristic, and ${\Bbb N}$ is the set of natural numbers. Let $L$ be a locally finite-dimensional (or {\em locally finite}, for brevity) Lie algebra. Assume for simplicity that $L$ has countable dimension. Then one can choose finite-dimensional subalgebras $L_i$ ($i\in{\Bbb N}$) of $L$ in such a way that $L_{i}\subset L_{i+1}$ for all $i$ and $L=\bigcup_{i\in{\Bbb N}}L_i$.The set $\{L_i\mid i\in{\Bbb N}\}$ is called a{\em local system} of $L$. Assume that $L$ is simple. Then all $L_i$ can be chosen to be perfect (that is, $[L_i,L_i]=L_i$). We shall call such local systems {\em perfect}. Let $S_i=S_i^1\oplus\ldots\oplus S_i^{n_i}$ be a Levi subalgebra of $L_i$, where $S_i^1,\ldots,S_i^{n_i}$ are simple components of $S_i$, and let $V_i^k$ be the standard $S_i^k$-module. Since $L_i$ is perfect, for every $k$ there exists a unique irreducible $L_i$-module${\cal V}_i^k$ such that the restriction ${\cal V}_i^k{\downarrow} S_i^k$ is isomorphic to $V_i^k$. An embedding $L_i\to L_{i+1}$ is called {\em diagonal} if, for each $l$, any non-trivial composition factor of the restriction ${\cal V}_{i+1}^l{\downarrow} L_i$ is isomorphic to ${\cal V}_i^k$ or its dual ${\cal V}_i^{k^*}$ for some $k$. We prove that a simple locally finite Lie algebra $L$ can be embedded into a locally finite associative algebra if and only if there exists a perfect local system $\{L_i\}_{i\in{\Bbb N}}$ of $L$ such that all embeddings $L_i\to L_{i+1}$ are diagonal. Note that this result can be considered as a version of Ado's theorem for simple locally finite Lie algebras. Let $V$ be a vector space. An element $x\in \frak{gl}(V)$ is called {\em finitary}if $\dim xV<\infty$. The finitary transformations of $V$ form an ideal $\frak{fgl}(V)$ of $\frak{gl}(V)$, and any subalgebra of $\frak{fgl}(V)$ is called a {\em finitary Lie algebra}. We classify all finitary simple Lie algebras of countable dimension. It turns out that there are only three: $\frak{sl}_\infty(F)$, $\frak{so}_\infty(F)$, and $\frak{sp}_\infty(F)$. The author has classified all finitary simple Lie algebras and will describe this in a subsequent paper.The analogous problem in group theory has been recently solved by J.I. Hall. He classified simple locally finite groups of finitary linear transformations.

1991 Mathematics Subject Classification: 17B35, 17B65.