Twisted K-theory for the orbifold \([\ast /G]\) (Q372518)

Let \(G\) be a finite group. The purpose of this paper is twofold: (a) to present an explicit ring isomorphism between the representation ring \(R(D^\omega(G))\) of the twisted Drinfeld double \(D^\omega(G)\) and the twisted orbifold \(K\)-theory \({ }^\omega K_{\text{orb}}([*/G])\) of the orbifold \([*/G]\) for any 3-cocycle \(\omega\) on \(G\) with values in \(S^1\), and (b) to examine a relation between the twisted orbifold \(K\)-theories for \([*/H])\) and \([*/(\mathbb{Z}_p)^n]\) where \(H\) is an extraspecial group of order \(p^n\) (\(p\) a prime).NEWLINENEWLINEHere \(D^\omega(G)\) is a quasi-Hopf algebra whose underlying vector space is \((\mathbb{C}G)^*\otimes (\mathbb{C}G)\). If \(g \in G\) let \(\delta_g\) be the function such that \(\delta_g(h)=1\) if \(h=g\) and 0 otherwise and write \(\delta_g\bar{x}\) for \(\delta_g\otimes\bar{x}\). Then its product and coproduct are given by \((\delta_g\bar{x})(\delta_h\bar{y})=\omega_g(x, y) \delta_g\delta_{xhx^{-1}}\overline{xy})\) and \(\Delta(\delta_g\bar{x})=\bigoplus_{h \in G}\gamma_x(h, h^{-1}g) (\delta_h\bar{x})\otimes(\delta_{h^{-1}g}\bar{x})\) resprectively, where both \(\omega_g\) and \(\gamma_x\) denote functions \(G\times G \to S^1\) induced by \(\omega\); in particular, the former one denotes the image of \(\omega\) via the inverse transgression map of \(g \in G\). Let \(C(g)\) denote the centralizer of \(g \in G\) and \(R_{\omega_g}(C(g))\) denote its Grotherndieck ring of classes of projective representations \((\rho, V)\), that is, a map \(\rho : C(g) \to GL(V, \mathbb{C})\) satisfying \(\rho(x)\rho(y)=\omega_g(x, y)\rho(xy)\). Firstly it is shown that \(R(D^\omega(G))\) is additively isomorphic to \(\bigoplus_{g \in C} R_{\omega_g}(C(g))\) where \(C\) is a set of representatives of the conjugacy classes in \(G\). On the other hand, we know that by definition \({ }^\omega K_{\text{orb}}([*/G]) =\bigoplus_{g \in C}{ }^{\omega_g} K_{C(g)}(*)\) and in this case \({ }^{\omega_g} K_{C(g)}(*)\) is just \(R_{\omega_g}(C(g))\). Combining these two facts we see that there is an additive isomorphism between \(R(D^\omega(G))\) and \({ }^\omega K_{\text{orb}}([*/G])\). In conclusion the authors prove that this additive isomorphism is in fact a ring isomorphism, which accomplishes (a). Based on this result, (b) is performed.