2010-09-24

Henri Moscovici, Ohio State University

Hopf algebras and Lie pseudogroups

The local symmetries of the classical geometric structures form the (primitive) Lie pseudogroups classified by E. Cartan. By linearization, to each such pseudogroup it corresponds a Lie algebra of formal vector fields, whose (Gelfand-Fuks) cohomology encapsulates the universal characteristic classes of foliations with corresponding transverse holonomy. The fact that the very same classes are responsible for the computation of indices of transversely elliptic operators on general foliations was revealed in work of A. Connes and myself, but only after we constructed a Hopf algebra whose cyclic cohomology turned out to deliver the very same classes in "readable" form. The theme of my talk will be a general construction, obtained in recent joint work with B. Rangipour, of such Hopf algebras of "transverse quantum symmetries" associated to primitive Lie pseudogroups.