104. 27/01/2025 Roman Stasenko (HSE University, Russia)

Title: 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 Φ: S → Aut(g) an S-structure on the Lie algebra g. S-structures were previously investigated 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 Φ 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 g be a Lie algebra with SL_2-structure and the map ρ:g→gl(U) be a linear representation of g. The homomorphism Φ:S→GL(U) is called an SL_2-structure on the Lie g-module U if Φ(s)ρ(\xi)u =ρ(Φ(s)\xi)\Psi(s)u. This construction 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 g-modules for simple Lie algebras.
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