Trace as an alternative decategorification functor

Categorification is a process of lifting structures to a higher categorical level. The original structure can then be recovered by means of the so-called "decategorification" functor. Algebras are typically categorified to additive categories with additional structure and decategorification is usually given by the (split) Grothendieck group. In this expository article we study an alternative decategorification functor given by the trace or the zeroth Hochschild--Mitchell homology. We show that this form of decategorification endows any 2-representation of the categorified quantum sl(n) with an action of the current algebra U(sl(n)[t]) on its center.Comment: 47 pages with tikz figures. arXiv admin note: text overlap with arXiv:1405.5920 by other author

Lectures on graded differential algebras and noncommutative geometry

These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments.Comment: 71 pages; minor typo correction

The algebra of Chern-Simons classes, the Poisson bracket on it, and the action of the gauge group

Developing ideas of the formal geometry [G1], [GKF], and ideas based on combinatorial formulas for characteristic classes we introduce the algebraic structure modeling N connections on the vector bundle over an oriented manifold. First we construct a graded free associative algebra A with a differential d. Then we go to the space V of cyclic words of A. Certain elements of V correspond to the secondary characteristic classes associated to k connections. That construction allows us to give easily the explicit formulas for some known secondary classes and to construct the new ones. Space V has new operations: it is a graded Lie algebra with respect to the Poisson bracket. We write how i-th differential and i-th homotopy operator in the algebra are connected with this bracket. There is an analogy between our algebra and the Kontsevich version of the noncommutative symplectic geometry. We consider then an algebraic model of the action of the gauge group. We describe how elements of our algebra corresponding to the secondary characteristic classes change under this action. 0. Introduction. The problem of localization of topological invariants by the methods of formal geometr