On the GL(n)-module structure of Lie nilpotent associative relatively free algebras

Elitza Hristova (Institute of Mathematics and Informatics, Bulgaria)

28-Aug-2023, 15:00-16:00 (7 months ago)

Abstract: Let $K\langle X\rangle$ denote the free associative algebra generated by a finite set $X$ with n elements over a field $K$ of characteristic 0. Let $I_p$ denote the two-sided associative ideal in $K\langle X\rangle$ generated by all commutators of length $p$, where $p$ is an arbitrary positive integer greater than 1. The group ${\rm GL(n)}$ acts in a natural way on the quotient $K\langle X\rangle/I_p$ and the ${\rm GL(n)}$-module structure of $K\langle X\rangle/I_p$ is known for $p=2,3,4,5$. In this talk, we give some results on the ${\rm GL}(n)$-module structure of $K\langle X\rangle/I_p$ for any $p$. More precisely, we give a bound on the values of the highest weights of irreducible ${\rm GL}(n)$-modules which appear in the decomposition of $K\langle X\rangle/I_p$. We discuss also applications of these results related to the algebras of G-invariants in $K\langle X\rangle/I_p$, where G is one of the classical ${\rm GL}(n)$-subgroups ${\rm SL}(n)$, ${\rm O}(n)$, ${\rm SO}(n)$, or ${\rm Sp}(2k)$ (for $n=2k$).

quantum algebrarings and algebras

Audience: researchers in the topic


European Non-Associative Algebra Seminar

Organizers: Ivan Kaygorodov*, Salvatore Siciliano, Mykola Khrypchenko, Jobir Adashev
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