Depth one extensions of semisimple algebras and Hopf subalgebras

TL;DR

This paper characterizes depth one extensions of semisimple algebras and Hopf algebras, relating them to their centers and extending known results from group algebra extensions.

Contribution

It provides a complete characterization of depth one extensions for semisimple algebras and Hopf algebras, generalizing previous results for group algebras.

Findings

Depth one extensions are characterized by their centers.

Results extend to semisimple Hopf algebras.

Analogous to finite group algebra extension results.

Abstract

An extension of $k$-algebras $B \subset A$ is said to have depth one if there exists a positive integer $n$ such that $ A$ is a direct summand of $ B^n$ in $_B\mtr{Mod}_B$. Depth one extensions of semisimple algebras are completely characterized in terms of their centers. For extensions of semisimple Hopf algebras our results are similar to those obtained for finite group algebra extensions in \cite{BKone}.