Subalgebra depth and double crossed products

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. $\mathbf{Subject}$ $\mathbf{classification: }$ $20C05$, $20G05$, $16W30$, $17B37$, $13E10$.