On isotypic decompositions for non-semisimple Hopf algebras

TL;DR

This paper explores the isotypic decomposition of regular modules in non-semisimple finite-dimensional Hopf algebras with the Chevalley property, providing explicit formulas for idempotents and demonstrating their completeness in many cases.

Contribution

It extends the understanding of isotypic decompositions to non-semisimple Hopf algebras with the Chevalley property, offering explicit formulas involving the dual Hopf algebra's regular character.

Findings

Explicit idempotent formulas for Hopf algebras with the Chevalley property

Complete orthogonal idempotent sets for a large class of Hopf algebras

Illustrative example emphasizing the importance of the Chevalley property

Abstract

In this paper we study the isotypic decomposition of the regular module of a finite-dimensional Hopf algebra over an algebraically closed field of characteristic zero. For a semisimple Hopf algebra, the idempotents realizing the isotypic decomposition can be explicitly expressed in terms of characters and the Haar integral. In this paper we investigate Hopf algebras with the Chevalley property, which are not necessarily semisimple. We find explicit expressions for idempotents in terms of Hopf-algebraic data, where the Haar integral is replaced by the regular character of the dual Hopf algebra. For a large class of Hopf algebras, these are shown to form a complete set of orthogonal idempotents. We give an example which illustrates that the Chevalley property is crucial.