BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Oksana Yakimova (University of Jena) DTSTART:20210304T195000Z DTEND:20210304T205000Z DTSTAMP:20260423T010005Z UID:GPRTatNU/18 DESCRIPTION:Title: Symmetrisation and the Feigin-Frenkel centre\nby Oksana Yakimova (Un iversity of Jena) as part of Geometry\, Physics\, and Representation Theor y Seminar\n\n\nAbstract\nLet $G$ be a complex reductive group\, set $\\mat hfrak g={\\mathrm{Lie\\\,}}G$. The algebra ${\\mathcal S}(\\mathfrak g)^{\ \mathfrak g}$ of symmetric $\\mathfrak g$-invariants and the centre ${\\ma thcal Z}(\\mathfrak g)$ of the enveloping algebra ${\\mathcal U}(\\mathfra k g)$ are polynomial rings in ${\\mathrm{rk\\\,}}\\mathfrak g$ generators. There are several isomorphisms between them\, including the symmetrisatio n map $\\varpi$\, which exists also for the Lie algebras $\\mathfrak q$ wi th $\\dim\\mathfrak q=\\infty$.\n\nHowever\, in the infinite dimensional c ase\, one may need to complete ${\\mathcal U}(\\mathfrak q)$ in order to r eplace ${\\mathcal Z}(\\mathfrak q)$ with an interesting related object. R oughly speaking\, the Feigin-Frenkel centre arises as a result of such com pletion in case of an affine Kac-Moody algebra. From 1982 until 2006\, thi s algebra existed as an intriguing black box with many applications. Then explicit formulas for its elements appeared first in type ${\\sf A}$\, lat er in all other classical types\, and it was discovered that the FF-centre is the centraliser of the quadratic Casimir element.\n\nWe will discuss t he type-free role of the symmetrisation map in the description of the FF-c entre and present new explicit formulas for its generators in types ${\\sf B}$\, ${\\sf C}$\, ${\\sf D}$\, and ${\\sf G}_2$. One of our main technic al tools is a certain map from ${\\mathcal S}^{k}(\\mathfrak g)$ to $\\Lam bda^2\\mathfrak g \\otimes {\\mathcal S}^{k-3}(\\mathfrak g)$.\n LOCATION:https://researchseminars.org/talk/GPRTatNU/18/ END:VEVENT END:VCALENDAR