EVENT DETAILS AND ABSTRACT

Title: Noncommutative Cartan calculus
Speaker: Boris Tsygan
Speaker Info: Northwestern University
Brief Description:
Special Note: This is the first talk in the continuation of the Tuesday afternoon seminar last quarter on Gauss-Manin connections in algebraic and noncommutative geometry. The seminar is from 3pm to 5pm, with a break for tea.
Abstract:

In calculus on manifolds, vector fields act on forms by Lie derivatives and also by contractions, and the actions satisfy the Cartan formulas. In noncommutative geometry, a manifold is replaced by an associative algebra, vector fields are replaced by derivations of this algebra, and forms are replaced by Hochschild chains. Derivations still act by Lie derivatives and contractions but the Cartan formulas change. Over the rationals, they can be restored, in a more relaxed (L_∞) sense. This allows to construct a flat super connection on the periodic cyclic complex of a family of algebras; this generalizes Getzler’s Gauss-Manin connection.

If the algebra is commutative, we get two algebraic structures described by Cartan formulas: one on forms and the other on Hochschild chains. I will explain how the Hochschild-Konstant-Rosenberg map from chains to forms intertwines the two structures.