Finite dimensional Frobenius–Perron algebras

Abstract

Motivated by the Frobenius–Perron (FP) theory of endofunctors of a [Formula: see text]-linear category, we introduce the notion of a FP algebra. This notion can be seen as a generalization of a fusion algebra, and is related to the PP theory of endofunctors of the categories of finite dimensional representations of Taft algebras. Our main result is to classify such algebras with small dimensions according to their FP dimension vectors. Moreover, we introduce the notion of a FP Hopf algebra which can be used to categorify a FP algebra via the Green ring of the Hopf algebra.