The ring structure for equivariant twisted K-theory

We prove, under some mild conditions, that the equivariant twisted K-theory group of a crossed module admits a ring structure if the twisting 2-cocycle is 2multiplicative. We also give an explicit construction of the transgression map T1 : H ∗(Γ • ;A) → H((N ⋊ Γ) • ;A) for any crossed module N → Γ and prove that any element in the image is ∞-multiplicative. As a consequence, we prove that, under some mild conditions, for a crossed module N → Γ and any e ∈ Ž(Γ • ;S), that the equivariant twisted K-theory groupK∗ e,Γ(N) admits a ring structure. As an application, we prove that for a compact, connected and simply connected Lie group G, the equivariant twisted K-theory group K∗ [c],G(G) is endowed with a canonical ring structureK [c],G(G)⊗K j+d [c],G(G) → K i+j+d [c],G (G), where d = dimG and [c] ∈ H ((G⋊G) • ;S).