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Register a new userCategorified Chern character and cyclic cohomology
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Authors
Ilya Shapiro
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We examine Hopf cyclic cohomology in the same context as the analysis of the geometry of loop spaces $LX$ in derived algebraic geometry and the resulting close relationship between $S^1$-equivariant quasi-coherent sheaves on $LX$ and $D_X$-modules. Furthermore, the Hopf setting serves as a toy case for the categorification of Chern character theory. More precisely, this examination naturally leads to a definition of mixed anti-Yetter-Drinfeld contramodules which reduces to that of the usual mixed complexes for the trivial Hopf algebra and generalizes the notion of stable anti-Yetter-Drinfeld contramodules that have thus far served as the coefficients for Hopf-cyclic theories. The cohomology is then obtained as a $Hom$ in this dg-category between a Chern character object associated to an algebra and an arbitrary coefficient mixed anti-Yetter-Drinfeld contramodule.
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This is the first one in a series of two papers on the continuation of our study in cup products in Hopf cyclic cohomology. In this note we construct cyclic cocycles of algebras out of Hopf cyclic cocycles of algebras and coalgebras. In the next paper we consider producing Hopf cyclic cocycle from equivariant Hopf cyclic cocycles. Our approach in both situations is based on (co)cyclic modules and bi(co)cyclic modules together with Eilenberg-Zilber theorem which is different from the old definition of cup products defined via traces and cotraces on DG algebras and coalgebras.
We give explicit formulae for the continuous Hochschild and cyclic homology and cohomology of certain topological algebras. To this end we show that, for a continuous morphism $phi: Xto Y$ of complexes of complete nuclear $DF$-spaces, the isomorphism of cohomology groups $H^n(phi): H^n(X) to H^n(Y)$ is automatically topological. The continuous cyclic-type homology and cohomology are described up to topological isomorphism for the following classes of biprojective $hat{otimes}$-algebras: the tensor algebra $E hat{otimes} F$ generated by the duality $(E, F, < cdot, cdot >)$ for nuclear Frechet spaces $E$ and $F$ or for nuclear $DF$-spaces $E$ and $F$; nuclear biprojective K{o}the algebras $lambda(P)$ which are Frechet spaces or $DF$-spaces; the algebra of distributions $mathcal{E}^*(G)$ on a compact Lie group $G$.
We define the $m$th Veronese power of a weight graded operad $mathcal{P}$ to be its suboperad $mathcal{P}^{[m]}$ generated by operations of weight $m$. It turns out that, unlike Veronese powers of associative algebras, homological properties of operads are, in general, not improved by this construction. However, under some technical conditions, Veronese powers of quadratic Koszul operads are meaningful in the context of the Koszul duality theory. Indeed, we show that in many important cases the operads $mathcal{P}^{[m]}$ are related by Koszul duality to operads describing strongly homotopy algebras with only one nontrivial operation. Our theory has immediate applications to objects as Lie $k$-algebras and Lie triple systems. In the case of Lie $k$-algebras, we also discuss a similarly looking ungraded construction which is frequently used in the literature. We establish that the corresponding operad does not possess good homotopy properties, and that it leads to a very simple example of a non-Koszul quadratic operad for which the Ginzburg--Kapranov power series test is inconclusive.
We present a new approach to cyclic homology that does not involve the Connes differential and is based on a `noncommutative equivariant de Rham complex of an associative algebra. The differential in that complex is a sum of the Karoubi-de Rham differential, which replaces the Connes differential, and another operation analogous to contraction with a vector field. As a byproduct, we give a simple explicit construction of the Gauss-Manin connection, introduced earlier by E. Getzler, on the relative cyclic homology of a flat family of associative algebras over a central base ring. We introduce and study `free-product deformations of an associative algebra, a new type of deformation over a not necessarily commutative base ring. Natural examples of free-product deformations arise from preprojective algebras and group algebras for compact surface groups.
Let $X$ be a compact Hausdorff space, let $Gamma$ be a discrete group that acts continuously on $X$ from the right, define $widetilde{X} = {(x,gamma) in X times Gamma : xcdotgamma= x}$, and let $Gamma$ act on $widetilde{X}$ via the formula $(x,gamma)cdotalpha = (xcdotalpha, alpha^{-1}gammaalpha)$. Results of P. Baum and A. Connes, along with facts about the Chern character, imply that $K^i_Gamma(X) otimes mathbb{C} cong K^i(widetilde{X}slashGamma) otimes mathbb{C}$ for $i = 0, -1$. In this note, we present an example where the groups $K^i_Gamma(X)$ and $K^i(widetilde{X}slashGamma)$ are not isomorphic.
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