Algebraic quotient modules and subgroup depth

TL;DR

This paper explores algebraic and module-theoretic characterizations of subgroup depth in Hopf algebras, establishing new equivalences and conditions, and applies these results to finite groups and their Drinfeld doubles.

Contribution

It introduces algebraic criteria for subgroup depth in Hopf algebras and relates it to module coalgebras, extending previous work and providing new applications.

Findings

Finite depth corresponds to algebraic elements in the Green ring.

Subgroup depth of corefree quotients lies within a one-unit interval.

Minimum depth of a group algebra in the Drinfeld double is an odd integer.

Abstract

In arXiv:1210.3178 it was shown that subgroup depth may be computed from the permutation module of the left or right cosets: this holds more generally for a Hopf subalgebra, from which we note in this paper that finite depth of a Hopf subalgebra R < H is equivalent to the H-module coalgebra Q = H/R^+H representing an algebraic element in the Green ring of H or R. This approach shows that subgroup depth and the subgroup depth of the corefree quotient lie in the same closed interval of length one. We also establish a previous claim that the problem of determining if R has finite depth in H is equivalent to determining if H has finite depth in its cross product Q* # H. A necessary condition is obtained for finite depth from stabilization of a descending chain of annihilator ideals of tensor powers of Q. As an application of these topics to a centerless finite group G, we prove that the minimum depth of its group complex algebra in the Drinfeld double D(G) is an odd integer, which determines the least tensor power of the adjoint representation Q that is a faithful complex G-module.