Root of unity quantum cluster algebras and discriminants

Abstract: We will define the notion of a root of unity quantum cluster algebra, which is not necessarily a specialization of a quantum cluster algebra. Through these algebras, we connect the subjects of cluster algebras and discriminants. Motivation for discriminants will be given in terms of their applications to representation theory. We show that the root of unity quantum cluster algebras are polynomial identity algebras, and we identify a large canonical central subalgebra. This central subalgebra is shown to be isomorphic to the underlying classical cluster algebra of geometric type. These central subalgebras can be thought of as a generalization of De Concini-Kac-Procesi's canonical central subalgebras for quantum groups at roots of unity. In particular, we recover their structure in the case of quantum Schubert cells. We prove a general theorem on the form of discriminants, which is given as a product of frozen cluster variables. From this we derive specific formulas in examples, such as for all root of unity quantum Schubert cells for any symmetrizable Kac-Moody algebra. This is joint work with Bach Nguyen and Milen Yakimov.

mathematical physicscommutative algebraalgebraic geometrycombinatoricsquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

Online Cluster Algebra Seminar (OCAS)