On the center of the small quantum group

TL;DR

This paper explores the structure of the center of the small quantum group at roots of unity, revealing a detailed decomposition involving the quantum Fourier transform and identifying the center's size relative to known subalgebras.

Contribution

It introduces a new subalgebra Z + F(Z) using the quantum Fourier transform, describing its structure and relation to the entire center of the small quantum group.

Findings

The subalgebra Z + F(Z) contains Z and describes the center's structure.

For sl_2, the center equals Z + F(Z); for other cases, it is larger.

The intersection Z ∩ F(Z) matches the annihilator of Z's radical.

Abstract

Using the quantum Fourier transform F, we describe the block decomposition and multiplicative structure of a subalgebra Z + F(Z) in the center of the small quantum group u_l at a root of unity. It contains the previously known subalgebra Z, which is isomorphic to the algebra of characters of finite dimensional modules over u_l. We prove that the intersection $Z \cap F(Z)$ coincides with the annihilator of the radical of Z. The whole center of u_l coincides with the obtained subalgebra Z + F(Z) in case of sl_2, and is bigger than that in general.