Hopf algebroids and secondary characteristic classes (Q960529)

Let \(U_n\) be the compact Lie group of \(n\times n\) unitary matrices and \(U^\delta _n\) the same group, but provided with the discrete topology. \(U^\delta _n \) acts by left multiplication on \(U_n\), and this defines a transformation groupoid \(U^\delta _n \ltimes U_n\). The paper studies the cyclic cohomology of the groupoid algebra \(C_c^\infty(U^\delta _n \ltimes U_n)\). There is a Hopf algebroid, \({\mathcal H}={\mathcal H} (U^\delta _n \ltimes U_n)\), naturally associated to the étale groupoid \(U^\delta _n \ltimes U_n\): different aspects of the cyclic cohomology of \({\mathcal H}\) are presented. In particular, the authors show that classes in the Hopf cyclic cohomology of \(\mathcal H\) can be used to define secondary characteristic classes of trivialized flat \(U_n\)-bundles. Applying Connes's index theorem, they find a link between their cyclic classes and the index classes in K-theory.