Hopf cyclic cohomology and Chern character of equivariant \(K\)-theories
Let \(H\) be a Hopf algebra with invertible antipode \(S\) over a field \({\mathbf k}\) of characteristic zero and let \(M\) be an anti-Yetter-Drinfeld module over \(H\). For the comultiplication of \(H\) and a coaction on \(M\) we write \(\Delta(h)=h^{(1)}\otimes h^{(2)}\) and \(\Delta_M(m) = m^{(-1)}\otimes m^{(0)}\). Then we have \(\Delta_M(mh) = S(h^{(3)})m^{(-1)}h^{(1)}\otimes m^{(0)}h^{(2)}\) and \(m^{(0)}m{(-1)}=m\). Let \(A\) be a Hopf module over \(H\) whose action satisfies \(h(ab)=h^{(1)}(a)h^{(2)}(b)\). These actions determine a cyclic module structure on \(C^H_n(A, M)=M\otimes_HA^{\otimes (n+1)}\), \(n \geq 0\), and so gives rise to the Hopf cyclic homology \(HC^H_*(A, M)\). Let \(V\) be a finite-dimensional \(H\)-module with a homomorphism \(\pi : H \to \mathrm{End}(V)\) and put \(B=\mathrm{End}(V)\otimes A\). If a linear map \(\lambda: H \to {\mathbf k}\) is given, then \(b \in B\) is called \(\lambda\)-stable if \(\lambda(h)b=\lambda(h^{(1)})h^{(2)}\cdot b\) holds. We consider an abelian semigroup consisting of all possible \(\lambda\)-stable idempotents \(p \in B\) where \(V\) runs over all the finite-dimensional \(H\)-modules and for its Grothendieck group we write \(K^{H, \lambda}(A)\) which is called the \(\lambda\)-stable \(K\)-theory. In this paper, using this \(K\)-theory the authors introduce the Chern character in the Hopf cyclic homology, which is given in the form \[ \mathrm{ch}^\lambda_n : K^{H, \lambda}(A) \to \Hom_{\mathbf k}(M, HC^H_{2n}(A, M)). \] In order to do this the authors construct two morphisms of cyclic modules \(HC_{2n}(B_\lambda) \to HC^H_{2n}(B, M)\) and \(HC^H_{2n}(B, M) \to HC^H_{2n}(A, M)\) where \(B_\lambda\) is the subalgebra of \(\lambda\)-stable elements of \(B\). Then we obtain \(\mathrm{ch}^\lambda_n\) by considering the composition of these two morphisms and the Chern character \[ \mathrm{ch}_n : K(B_\lambda) \to HC_{2n}(B_\lambda) \] in the ordinary cyclic homology. Moreover the authors introduce a Chern character \(\mathrm{ch}_G\) on a sheaf of the \(G\)-equivariant \(K\)-theory where \(G\) acts on itself by conjugation. Let \(X\) be a \(G\)-manifold and \(\mathcal{K}^0_G(X)\) denote the sheaf with stalk \(K^0_{G^g}(X^g)\) at \(g \in G\). Given a section \(\gamma\) of \(\mathcal{K}^0_G(X)\) over an invariant subset \(U \subset G\), \(\gamma(g)\) can be identified with a \(G^g\)-vector bundle \(\xi_g\) over \(X^g\). Then \(\mathrm{ch}_G(\gamma)\) is defined using a \(G^g\)-connection \(\nabla\) on \(\xi_g\).
scientific article; zbMATH DE number 6509522
scientific article