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 2-multiplicative. We also give an explicit construction of the transgression map $T_1:H^{\ast }(\mathrm{\Gamma };A)\to H^{\ast -1}((N⋊\mathrm{\Gamma };A)$ for any crossed module $N\to \mathrm{\Gamma }$ and prove that any element in the image is $\mathrm{\infty }$-multiplicative. As a consequence, we prove that, under some mild conditions, for a crossed module $N\to \text{gm}$ and any $e \in \check{Z}^3(\Gamma;S^1)$,thattheequivarianttwistedK-theorygroup$K^*{e,\Gamma}(N)$ 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 $K_{[c],G}^{\ast }(G)$ is endowed with a canonical ring structure $K^{i+d}{[c],G}(G)\otimes K^{j+d}{[c],G}(G)\to K^{i+j+d}{[c], G}(G)$,where$d=dim G$and$[c]\in H^2(G\rtimes G;S^1)$.