Root of unity quantum cluster algebras

Abstract: We will describe a theory of root of unity quantum cluster algebras, which cover as special cases the big quantum groups of De Concini, Kac and Process. All such algebras will be shown to be polynomial identity (PI) algebras. Inside each of them, we will construct a canonical central subalgebra which is proved to be isomorphic to the underlying cluster algebra. It is a far-reaching generalization of the De Concini-Kac-Procesi central subalgebras in big quantum groups and presents a general framework for studying the representation theory of quantum algebras at roots of unity by means of cluster algebras as the relevant data becomes (PI algebra, canonical central subalgebra)=(root of unity quantum cluster algebra, underlying cluster algebra). We will also present an explicit formula for the corresponding discriminants in this general setting that can be applied in many concrete situations of interest, such as the discriminants of all root of unity quantum unipotent cells for symmetrizable Kac-Moody algebras. This is a joint work with Bach Nguyen (Xavier Univ) and Kurt Trampel (Notre Dame Univ).

mathematical physicsalgebraic geometrydifferential geometrygeometric topologyoperator algebrasrepresentation theorysymplectic geometry

Audience: researchers in the topic

Geometry, Physics, and Representation Theory Seminar

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