The center of small quantum groups I: the principal block in type A

TL;DR

This paper introduces an algebraic method to compute the center of the principal block of small quantum groups for complex semisimple Lie algebras, providing explicit calculations for  and conjecturing a connection to Haiman's diagonal coinvariant algebra.

Contribution

It develops a new elementary algebraic approach to compute centers of small quantum groups and formulates a conjecture linking these centers to well-known algebraic structures.

Findings

Explicit computation of the center for  case.

Sketch of the  case evidence.

Conjecture relating centers to Haiman's diagonal coinvariant algebra.

Abstract

We develop an elementary algebraic method to compute the center of the principal block of a small quantum group associated with a complex semisimple Lie algebra at a root of unity. The exemplary case of $\mathfrak{sl}_3$ is computed explicitly, and further evidence of $\mathfrak{sl}_4$ is sketched. This allows us to formulate the conjecture that, as a bigraded vector space, the center of a regular block of the small quantum $\mathfrak{sl}_m$ at a root of unity is isomorphic to Haiman's diagonal coinvariant algebra for the symmetric group $S_{m}$.