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For finite groups $X$, $G$ and the right $G$-action on $X$ by group automorphisms, the non-balanced quantum double $D(X;G)$ is defined as the crossed product ${\left(ℂX^{\mathrm{op}}\right)}^{*}⋊ℂG$. We firstly prove that $D(X;G)$ is a finite-dimensional Hopf $C^{*}$-algebra. For any subgroup $H$ of $G$, $D(X;H)$ can be defined as a Hopf $C^{*}$-subalgebra of $D(X;G)$ in the natural way. Then there is a conditonal expectation from $D(X;G)$ onto $D(X;H)$ and the index is $[G;H]$. Moreover, we prove that an associated natural inclusion of non-balanced quantum doubles is the crossed product by the group algebra....

Let $G$ be a finite group and $H$ a subgroup. Denote by $D(G;H)$ (or $D\left(G\right)$) the crossed product of $C\left(G\right)$ and $ℂH$ (or $ℂG$) with respect to the adjoint action of the latter on the former. Consider the algebra $\langle D\left(G\right),e\rangle$ generated by $D\left(G\right)$ and $e$, where we regard $E$ as an idempotent operator $e$ on $D\left(G\right)$ for a certain conditional expectation $E$ of $D\left(G\right)$ onto $D(G;H)$. Let us call $\langle D\left(G\right),e\rangle$ the basic construction from the conditional expectation $E:D\left(G\right)\to D(G;H)$. The paper constructs a crossed product algebra $C(G/H\times G)⋊ℂG$, and proves that there is an algebra isomorphism between $\langle D\left(G\right),e\rangle$ and $C(G/H\times G)⋊ℂG$.