Hopf subalgebras and tensor powers of generalized permutation modules

TL;DR

This paper investigates the conditions under which the depth of a Hopf subalgebra in a finite-dimensional Hopf algebra is finite, using tensor powers of generalized permutation modules within a finite tensor category.

Contribution

It introduces criteria involving module properties and tensor powers that ensure the finiteness of Hopf subalgebra depth, extending understanding of module coalgebras and tensor categories.

Findings

Finite representation type implies finite depth.

Semisimple modules with certain properties have finite depth.

Depth bounds are derived from tensor algebra comparisons.

Abstract

By means of a certain module V and its tensor powers in a finite tensor category, we study a question of whether the depth of a Hopf subalgebra R of a finite-dimensional Hopf algebra H is finite. The module V is the counit representation induced from R to H, which is then a generalized permutation module, as well as a module coalgebra. We show that if in the subalgebra pair either Hopf algebra has finite representation type, or V is either semisimple with R* pointed, projective, or its tensor powers satisfy a Burnside ring formula over a finite set of Hopf subalgebras including R, then the depth of R in H is finite. One assigns a nonnegative integer depth to V, or any other H-module, by comparing the truncated tensor algebras of V in a finite tensor category and so obtains an upper and lower bound for depth of a Hopf subalgebra. For example, a relative Hopf restricted module has depth 1, and a permutation module of a corefree subgroup has depth less than the number of values assumed by its character.