Birational Normal Forms for Kac--Moody Algebras

Abstract: The classical Gel'fand--Kirillov conjecture posits that the quotient division algebra of an enveloping algebra is a Weyl skew field. The conjecture is known for finite-dimensional solvable Lie algebras, hence for Borel subalgebras of finite-dimensional semisimple Lie algebras. The conjecture also holds for their \(q\)-deformations at generic \(q\). In this talk, I will explain a Kac--Moody version of the birational problem, where the correct normal form is no longer a single Weyl algebra but a pair of Weyl-polynomial Borel models glued by a smash-biproduct structure. Our main result identifies the two Borel halves of a Kac--Moody algebra associated with a generalized Cartan matrix \(C\) of corank \(\ell\) with the birational model \(A_{n-\ell,n}\otimes k[t_1,\ldots,t_\ell]\), where \(A_{n-\ell,n}\) is the corresponding rectangular Weyl algebra. The proof goes through controlled Ore localizations and Cartan-bound generalized Weyl algebras, making the birational transformation explicit instead of passing directly to the maximal localization. The corank of \(C\) determines the number of residual polynomial variables. I will also explain how the same mechanism extends to Drinfeld--Jimbo quantizations by replacing Weyl algebras with their \(q\)-analogues.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar