Subalgebra depth and double crossed products

TL;DR

This paper investigates the depth of ring extensions involving factorized algebras, generalizing previous results for Hopf algebras and applying to double crossed products and Drinfeld doubles.

Contribution

It extends the concept of depth to generalised smash products and double crossed products, broadening understanding of algebraic structures in Hopf algebra theory.

Findings

Generalized depth results for smash products and quotient module coalgebras

New bounds for depth in double crossed product extensions

Application to the depth of Hopf algebra in its Drinfeld double

Abstract

In this paper we explore the concept of depth of a ring extension when the overall algebra factorises as a product of two subalgebras, in particular the case of finite dimensional Hopf algebras. As a result we generalise the results by Kadison and Young \cite{HKY} on depth of a Hopf algebra $H$ in its smash product with a finite dimensional left $H$-module algebra $A$, $A#H$ to the context of generalised smash products $Q^{*op}#_\psi H$ \cite{Bz1} where $Q$ is the quotient module coalgebra associated to the extension $R\subseteq H$ of finite dimensional Hopf algebras \cite{Ka2}\cite{HKY}\cite{H}. Moreover, following the construction of double crossed products in \cite{Ma} and \cite{Ma1} we use our result on factorisation algebras to get a general result on the depth of the extension of a Hopf algebra $H$ in its Drinfel\vtick d double $D(H)$. $\mathbf{Keywords : }$ Depth, Factorisation Algebra, Smash Product, Drinfel\vtick d Double, Double Crossed Product, Normal Extension. $\mathbf{Subject}$ $\mathbf{classification: }$ $20C05$, $20G05$, $16W30$, $17B37$, $13E10$.