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SUMMARY:Georgy Sharygin
DTSTART:20230315T162000Z
DTEND:20230315T180000Z
DTSTAMP:20260423T041623Z
UID:GDEq/85
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/GDEq/85/">Qu
 asiderivations and commutative subalgebras of the algebra $U\\mathfrak{gl}
 _n$</a>\nby Georgy Sharygin as part of Geometry of differential equations 
 seminar\n\nLecture held in room 303 of the Independent University of Mosco
 w.\n\nAbstract\nLet $\\mathfrak{gl}_n$ be the Lie algebra of $n\\times n$ 
 matrices over a characteristic zero field $\\Bbbk$ (one can take $\\Bbbk=\
 \mathbb R$ or $\\mathbb C$)\; let $S(\\mathfrak{gl}_n)$ be the Poisson alg
 ebra of polynomial functions on $\\mathfrak{gl}_n^*$\, and $U\\mathfrak{gl
 }_n$ the universal enveloping algebra of $\\mathfrak{gl}_n$. By Poincaré-
 Birkhoff-Witt theorem $S(\\mathfrak{gl}_n)$ is isomorphic to the graded al
 gebra $gr(U\\mathfrak{gl}_n)$\, associated with the order filtration on $U
 \\mathfrak{gl}_n$. Let $A\\subseteq S(\\mathfrak{gl}_n)$ be a Poisson-comm
 utative subalgebra\; one says that a commutative subalgebra $\\hat A\\subs
 eteq U\\mathfrak{gl}_n$ is a $\\textit{quantisation}$ of $A$\, if its imag
 e under the natural projection $U\\mathfrak{gl}_n\\to gr(U\\mathfrak{gl}_n
 )\\cong S(\\mathfrak{gl}_n)$ is equal to $A$.\n\nIn my talk I will speak a
 bout the so-called "argument shift" subalgebras $A=A_\\xi$ in $S(\\mathfra
 k{gl}_n)$\, generated by the iterated derivations of central elements in $
 S(\\mathfrak{gl}_n)$ by a constant vector field $\\xi$. There exist severa
 l ways to define a quantisation of $A_\\xi$\, most of them are related wit
 h the considerations of some infinite-dimensional Lie algebras. In my talk
  I will explain\, how one can construct such quantisation of $A_\\xi$ usin
 g as its generators iterated $\\textit{quasi-derivations}$ $\\hat\\xi$ of 
 $U\\mathfrak{gl}_n$. These operations are "quantisations" of the derivatio
 ns on $S(\\mathfrak{gl}_n)$ and verify an analog of the Leibniz rule. In f
 act\, I will show that iterated quasiderivation of certain generating elem
 ents in $U\\mathfrak{gl}_n$ are equal to the linear combinations of the el
 ements\, earlier constructed by Tarasov.\n
LOCATION:https://researchseminars.org/talk/GDEq/85/
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