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Hopf-cyclic homology and characteristic classes of equivariant K-theory

Le : 11/02/2015 10h30
Par : Georgi Sharygin (Université d'Etat de Moscou)
Lieu : I 001
Lien web :
Résumé : The notion of Hopf-cyclic homology sprung from a paper by Connes and Moscovici, in which they described the index of a pseudo-elliptic operator on a foliation in terms of a pairing of the Connes Chern character of a vector bundle and a certain class in cyclic cohomology of the convolution algebra $\mathcal A$ of the foliation. This latter class was shown to belong to the image of a characteristic mapping to $HC^*(\mathcal A)$ from the cyclic cohomology of a cyclic module, associated with certain Hopf algebra (the universal Hopf algebra $\mathcal H_n$ in their notation). It was later shown, that similar cyclic (co)homology, called the Hopf-cyclic (co)homology, can be constructed for arbitrary Hopf algebras, and even bialgebras, and that certain coefficient modules (the stable anti-Yetter-Drinfeld modules or SAYD modules for short) can be used in this case. The construction of characteristic map has also been extended to these new constructions, leading to the notion of characteristic pairing in Hopf-cyclic homology. In my talk I will explain the construction of Hopf-cyclic (co)homology with coefficients in SAYD module. I will only briefly talk about the characteristic pairing in this case. Instead I will give an "opposite" construction: as one knows, cyclic homology is the range of Connes Chern character from K-theory. So, the question is, if similar can be said about the Hopf-cyclic theory and what role is played by coefficients in this constructions. I will try to construct an appropriate K-theory ("with coefficients"), which would be equipped with a characteristic map into the Hopf-cyclic theory. More specifically, I will propose two alternative constructions of such K-theory: one purely algebraic, in which the "coefficients" come from a character on the algebra, and another one imitating the Block-Getzler's equivariant cohomology sheaf construction. They both give rise to characteristic maps of the type I described.