BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Atabey Kaygun (Istanbul Technical University\, Turkey) DTSTART:20260706T150000Z DTEND:20260706T160000Z DTSTAMP:20260714T041306Z UID:ENAAS/180 DESCRIPTION:Title: Birational Normal Forms for Kac--Moody Algebras\nby Atabey Kaygun (Ist anbul Technical University\, Turkey) as part of European Non-Associative A lgebra Seminar\n\n\nAbstract\nThe classical Gel'fand--Kirillov conjecture posits that the quotient division algebra of an enveloping algebra is a We yl skew field. The conjecture is known for finite-dimensional solvable Lie algebras\, hence for Borel subalgebras of finite-dimensional semisimple L ie algebras. The conjecture also holds for their \\(q\\)-deformations at g eneric \\(q\\). In this talk\, I will explain a Kac--Moody version of the birational problem\, where the correct normal form is no longer a single W eyl algebra but a pair of Weyl-polynomial Borel models glued by a smash-bi product structure. Our main result identifies the two Borel halves of a Ka c--Moody algebra associated with a generalized Cartan matrix \\(C\\) of co rank \\(\\ell\\) with the birational model \\(A_{n-\\ell\,n}\\otimes k[t_1 \,\\ldots\,t_\\ell]\\)\, where \\(A_{n-\\ell\,n}\\) is the corresponding r ectangular Weyl algebra. The proof goes through controlled Ore localizatio ns and Cartan-bound generalized Weyl algebras\, making the birational tran sformation explicit instead of passing directly to the maximal localizatio n. The corank of \\(C\\) determines the number of residual polynomial vari ables. I will also explain how the same mechanism extends to Drinfeld--Jim bo quantizations by replacing Weyl algebras with their \\(q\\)-analogues.\ n LOCATION:https://researchseminars.org/talk/ENAAS/180/ END:VEVENT END:VCALENDAR