MIT Infinite Dimensional Algebra Seminar
Friday, September 17, 2021 at 3:00pm to 5:00pm
Building 2, 1-135
182 MEMORIAL DR, Cambridge, MA 02139
Title: Root of unity quantum cluster algebras: discriminants, Cayley-Hamilton algebras, and Poisson orders
Abstract: We will describe a theory of root of unity quantum cluster algebras, which includes various families of algebras from Lie theory and topology. All such algebras will be shown to be maximal orders in central simple algebras. Inside each of them, we will construct a canonical central subalgebra which is isomorphic to the underlying cluster algebra. It is a far-reaching generalization of the De Concini-Kac-Procesi central subalgebras that play a fundamental role in the representation theory of quantum groups at roots of unity. An explicit formula for the corresponding discriminants will be presented. We will also show that all root of unity quantum cluster algebras have canonical structures of Cayley-Hamilton algebras (in the sense of Procesi) and Poisson orders (in the sense of De Concini-Kac-Procesi and Brown-Gordon). Their fully Azumaya loci will be shown to contain the underlying cluster A varieties. This is a joint work with Shengnan Huang, Thang Le, Bach Nguyen and Kurt Trampel.