Unrestricted quantum moduli algebras, II: Noetherianity and simple fraction rings at roots of $1$

Abstract

We prove that the unrestricted quantum moduli algebra of a punctured sphere and complex simple Lie algebra g is a finitely generated ring and a Noetherian ring, and that specializations at roots of unity of odd order l embed in a natural way in a central simple algebra of PI degree l (n−1)N−m , where N is the number of positive roots of g, m its rank, and n + 1 ≥ 3 the number of punctures.

Unrestricted Quantum Moduli Algebras. I. The Case of Punctured Spheres

“…In our context we use also tools, like filtrations, which are standard for quantum groups, and a version of the Hilbert-Nagata theorem in invariant theory. The similar result when g = 0 was obtained in [BR21]. The present genus g > 0 case is substantially more complicated.…”

“…In [11] and works in preparation we study the structure of the algebra L ϵ 0,n and its subalgebras (L ϵ 0,n ) Uϵ and L A 0,n…”

“…Let us discuss a bit how we prove Theorem 1. The main ideas of the proof are similar to those for L 0,n in [BR19,BR21]; however when g > 0 the presence of the algebra L 1,0 requires many non-trivial generalizations and new computations.…”

“…By using the Kashiwara-Lusztig theory of canonical basis, our proof that L g,n (g) is Noetherian can be adapted to show that L A g,n (g) is Noetherian as well, and moreover it is a finitely generated ring. Full details in the case g = 0 are given in [BR21].…”

“…Hence the algebras L 0,1 (H) and L 1,0 (H) play a special role and have to be examined first. The papers [BR19,BR21] were concerned with g = 0 and in particular with L 0,1 (H). Here we are interested in L g,n (H) for arbitrary g. So we start with L 1,0 (H) in §3.…”