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

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 ℂH (or ℂG) with respect to the adjoint action of the latter on the former. Consider the algebra 〈D(G), e〉 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 〈D(G), e〉 the basic construction from the conditional expectation E: D(G) → D(G; H). The paper constructs a crossed product algebra C(G/H ×G)⋊ℂG, and proves that there is an algebra isomorphism between 〈D(G), e〉 and C(G/H×G)⋊ℂG.