Algebraic quotient modules and subgroup depth

Abstract

In Kadison J Pure Appl Alg 218:367–380, (2014) 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 $R \subseteq H$ 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 smash 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 $\mathbb {C}\,$-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 $\mathbb {C}\,G$-module.