Small semisimple subalgebras of semisimple Lie algebras

Abstract

The main goal of this paper is to prove the following theorem: Let $\mathfrak{k}$ be an ${\mathfrak{sl}}_2$-subalgebra of a semisimple Lie algebra $\mathfrak{g}$, none of whose simple factors is of type $A1$. Then there exists a positive integer $b(\mathfrak{k},\mathfrak{g})$, such that for every irreducible finite dimensional $\mathfrak{g}$-module $V$, there exists an injection of $\mathfrak{k}$-modules $W\to V$, where $W$ is an irreducible $\mathfrak{k}$-module of dimension less than $b(\mathfrak{k},\mathfrak{g})$. This result was announced in math.RT/0310140.

16 pages