The basic construction from the conditional expectation on the quantum double of a finite group

摘要

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