Indecomposable Hopf $*$-algebra representations with invariant inner product

TL;DR

This paper extends Araki's 1985 result to Hopf $*$-algebras, characterizing invariant inner products on indecomposable representations of small quantum groups and generalized Taft algebras.

Contribution

It generalizes the classification of invariant inner products from group representations to Hopf $*$-algebras and provides explicit characterizations for specific quantum and Taft algebras.

Findings

Invariant inner products are characterized for small quantum groups at roots of unity.

Invariant inner products are characterized for indecomposable representations of generalized Taft algebras.

The results extend previous work on group representations to a broader algebraic setting.

Abstract

We generalize a result of Araki (1985) on indecomposable group representations with invariant (necessarily indefinite) inner product and irreducible subrepresentation to Hopf $*$-algebras. Moreover, we characterize invariant inner products on the projective indecomposable representations of small quantum groups $U_qsl(2)$ at odd roots of unity and on the indecomposable representations of generalized Taft algebras $H_{n,d}(q)$.