On the Hopf (co)center of a Hopf algebra

TL;DR

This paper explores the structures of Hopf center and cocenter in Hopf algebras, establishing their properties through extension theory and computing them for quantum groups and related algebraic structures.

Contribution

It introduces the extension-theoretic approach to Hopf (co)centers and computes these for key quantum algebra examples, advancing understanding of their algebraic structure.

Findings

Hopf center and cocenter form exact sequences of Hopf algebras.

Exact sequences satisfy faithful (co)flatness conditions.

Explicit computations for quantum groups U_q(g) and Pol(G_q).

Abstract

The notion of Hopf center and Hopf cocenter of a Hopf algebra is investigated by the extension theory of Hopf algebras. We prove that each of them yields an exact sequence of Hopf algebras. Moreover the exact sequences are shown to satisfy the faithful (co)flatness condition. Hopf center and cocenter are computed for $\mathsf{U}_q(\mathfrak{g})$ and the Hopf algebra $\textrm{Pol}(\mathbb{G}_q)$, where $\mathbb{G}_q$ is the Drinfeld-Jimbo quantization of a compact semisimple simply connected Lie group $\mathbb{G}$ and $\mathfrak{g}$ is a simple complex Lie algebra.