Minimum depth of factorization algebra extensions

Abstract

In this paper we study the minimum depth of a subalgebra embedded in a factorization algebra, and outline how subring depth, in this context, is related to module depth of the regular left module representation of the given subalgebra, within the appropriate module ring. As a consequence, we produce specific results for subring depth of a Hopf subalgebra in its Drinfel'd double. Moreover we study a necessary and sufficient condition for normality of a Hopf algebra within a double cross product context, which is equivalent to depth 2, as it is well known by a result of Kadison. Using the sufficient condition, we then prove some results regarding minimum depth 2 for Drinfel'd double Hopf subalgebra pairs, particularly in the case of finite group algebras. Finally, we provide formulas for the centralizer of a normal Hopf subalgebra in a double cross product scenario.