Short $SL_2$-structures on simple Lie algebras and Lie's modules

Abstract: Let $S$ be an arbitrary reductive algebraic group. Let's call a homomorphism $\Phi:S\rightarrow\operatorname{Aut}(\mathfrak{g})$ an {\it $S$-structure on the Lie algebra $\mathfrak{g}$}. $S$-structures were previously invetigated by various authors, including E.B. Vinberg. The talk deals with $SL_2$-structures. Let's call the $SL_2$-structure short if the representation $\Phi$ of the group $SL_2$ decomposes into irreducible representations of dimensions 1, 2 and 3. If we consider irreducible representations of dimensions only 1 and 3, we get the well-known Tits-Kantor-Koeher construction, which establishes a one-to-one correspondence between simple Jordan algebras and simple Lie algebras of a certain type. Similarly to the Tits–Kantor–Koeher theorem, in the case of short $SL_2$-structures, there is a one-to-one correspondence between simple Lie algebras with such a structure and the so-called simple symplectic Lie-Jordan structures. Let $\mathfrak{g}$ be a Lie algebra with $SL_2$-structure and the map $\rho:\mathfrak{g}\rightarrow\mathfrak{gl}(U)$ be linear representation of $\mathfrak{g}$. The homophism $\Psi:S\rightarrow GL(U)$ is called a $SL_2$-structure on the Lie $\mathfrak{g}$-module $U$ if $$\Psi(s)\rho(\xi)u =\rho(\Phi(s)\xi)\Psi(s)u,\quad\forall s\in S, \xi\in\mathfrak{g}, u\in U.$$ This constrution has interesting applications to the representation theory of Jordan algebras, which will be discussed during the talk. We will also present a complete classification of irreducible short $\mathfrak{g}$-modules for simple Lie algebras.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar