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SUMMARY:Roman Stasenko (HSE University\, Russia)
DTSTART:20250127T150000Z
DTEND:20250127T160000Z
DTSTAMP:20260423T035604Z
UID:ENAAS/108
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/ENAAS/108/">
 Short $SL_2$-structures on simple Lie algebras and Lie's modules</a>\nby R
 oman Stasenko (HSE University\, Russia) as part of European Non-Associativ
 e Algebra Seminar\n\n\nAbstract\nLet $S$ be an arbitrary reductive algebra
 ic group. Let's call  a homomorphism $\\Phi:S\\rightarrow\\operatorname{Au
 t}(\\mathfrak{g})$ an {\\it $S$-structure on the Lie algebra $\\mathfrak{g
 }$}. $S$-structures were previously invetigated by various authors\, inclu
 ding 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 $SL_2$ de
 composes 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-t
 o-one correspondence between simple Jordan algebras and simple Lie algebra
 s 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 simp
 le symplectic Lie-Jordan structures. Let  $\\mathfrak{g}$ be a Lie algebra
  with $SL_2$-structure and the map $\\rho:\\mathfrak{g}\\rightarrow\\mathf
 rak{gl}(U)$ be linear representation of $\\mathfrak{g}$. The homophism $\\
 Psi:S\\rightarrow GL(U)$ is called a $SL_2$-structure on the Lie $\\mathfr
 ak{g}$-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 construt
 ion has interesting applications to the representation theory of Jordan al
 gebras\, which will be discussed during the talk. We will also present a c
 omplete classification of irreducible short $\\mathfrak{g}$-modules for si
 mple Lie algebras.\n
LOCATION:https://researchseminars.org/talk/ENAAS/108/
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