The cyclic homology H C*(A) of an associative algebra with unit A over a field k of characteristic zero was introduced by A. Connes [C1], and extended to arbitrary commutative rings k in [LQ]. It comes equipped with a natural degree (-2) k-linear map S: HC*(A)~HC*_2(A). We will occasionally write S as SA to indicate dependence on the ring A; the map SA provides HC*(A) with a natural k[u] co-mod...

The cyclic homology of an exact category was defined by R. McCarthy [17] using the methods of F. Waldhausen [26]. McCarthy’s theory enjoys a number of desirable properties, the most basic being the extension property, i.e. the fact that when applied to the category of finitely generated projective modules over an algebra it specializes to the cyclic homology of the algebra. However, we show tha...

A new class of coefficients for the Hopf-cyclic homology of module algebras and coalgebras is introduced. These coefficients, termed stable anti-Yetter-Drinfeld contramodules, are both modules and contramodules of a Hopf algebra that satisfy certain compatibility conditions. 1. Introduction. It has been demonstrated in [8], [9] that the Hopf-cyclic homology developed by Connes and Moscovici [5]...

We present a brief introduction to Hochschild and cyclic homology designed for researchers interested in pseudodifferential operators and their applications to index theory, spectral invariants, and asymptotics.

We use some BRS techniques to construct Chern-Simons forms generalizing the Chern character of K1 groups in the Cuntz-Quillen description of cyclic homology. MSC91: 19D55, 81T13, 81T50

We show that Khovanov homology and Hochschild homology theories share common structure. In fact they overlap: Khovanov homology of a (2, n)-torus link can be interpreted as a Hochschild homology of the algebra underlining the Khovanov homology. In the classical case of Khovanov homology we prove the concrete connection. In the general case of Khovanov-Rozansky, sl(n), homology and their deforma...

We define a new cyclic module, dual to the Connes-Moscovici cyclic module, for Hopf algebras, and give a characteristric map for coactions of Hopf algebras. We also compute the resulting cyclic homology for cocommutative Hopf algebras, and some quantum groups.

We show that the reduced Hochschild homology of a DG open Frobenius algebra has the natural structure of a Batalin-Vilkovisky coalgebra, and the reduced cyclic homology has the natural structure of a gravity coalgebra. This gives an algebraic model for a Batalin-Vilkovisky coalgebra structure on the reduced homology of the free loop space of a simply connected closed oriented manifold, and a gr...

We consider a pairing producing various cyclic Hochschild cocycles, which led Alain Connes to cyclic cohomology. We are interested in geometrical meaning and homological properties of this pairing. We define a non-trivial pairing between the homology of a Lie-Rinehart (super-)algebra with coefficients in some partial traces and relative periodic cyclic homology. This pairing generalizes the ind...