BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Georgy Sharygin DTSTART:20210512T162000Z DTEND:20210512T180000Z DTSTAMP:20260423T010249Z UID:GDEq/40 DESCRIPTION:Title: Op erations on universal enveloping algebra and the "argument shift" method\nby Georgy Sharygin as part of Geometry of differential equations semin ar\n\nLecture held in room 303 or 304 of the Independent University of Mos cow.\n\nAbstract\nIf a vector field X is given on a Poisson manifold M suc h that the square of the Lie derivative in the X direction "kills" the Poi sson bivector\, then there is a well-known simple method of "shifting the argument" (along X) to construct a commutative subalgebra (with respect to the Poisson bracket) inside the algebra of functions on M. In a particula r case\, this method can be applied to the Poisson-Lie bracket on the symm etric algebra of an arbitrary Lie algebra and gives (according to a well-k nown result\, the proven Mishchenko-Fomenko conjecture) maximal commutativ e subalgebras in the symmetric algebra. However\, the lifting of these alg ebras to commutative subalgebras in the universal enveloping algebra\, alt hough possible\, is based on very nontrivial results from the theory of in finite-dimensional Lie algebras. In my talk\, I will describe partial resu lts that allow one to construct on the universal enveloping algebra of the algebra $gl_{n}$ the operators of "quasidifferentiation" and with thei r help\, in some cases\, construct a commutative subalgebra in $Ugl_{n}$ . I will also describe how\, in the general case\, this question is red uced to the combinatorial question of commuting a certain set of operators in tensor powers $\\mathbb {R} ^{n}$. The story is based on collaborat ions with Dmitry Gurevich\, Pavel Saponov and Ikeda Yasushi.\n LOCATION:https://researchseminars.org/talk/GDEq/40/ END:VEVENT END:VCALENDAR