Bialgebra Cyclic Homology with Coefficients, Part I

The cyclic (co)homology of Hopf algebras is defined by Connes and Moscovici [math.DG/9806109] and later extended by Khalkhali et.al [math.KT/0306288] to admit stable anti-Yetter-Drinfeld coefficient module/comodules. In this paper we will show that one can further extend the cyclic homology of Hopf algebras with coefficients non-trivially. The new homology, called the bialgebra cyclic homology, admits stable coefficient module/comodules, dropping the anti-Yetter-Drinfeld condition. This fact allows the new homology to use bialgebras, not just Hopf algebras. We will also give computations for bialgebra cyclic homology of the Hopf algebra of foliations of codimension $n$ and the quantum deformation of an arbitrary semi-simple Lie algebra with several stable but non-anti-Yetter-Drinfeld coefficients.Comment: 23 pages, LaTeX, no figures. v2: Corollary 3.9 and Corollary 3.14 were removed. v3: A more refined notion of stability is introduced and an explicit definition of aYD modules is give

Uniqueness of pairings in Hopf-cyclic cohomology

We show that all pairings defined in the literature extending Connes-Moscovici characteristic map in Hopf-cyclic cohomology are isomorphic as natural transformations of derived double functors.Comment: 16 pages; Section 3 is re-writte

Hopf--Hochschild (co)homology of module algebras

We define a version of Hochschild homology and cohomology suitable for a class of algebras admitting compatible actions of bialgebras, called module algebras. We show this (co)homology, called Hopf--Hochschild (co)homology, can also be defined as a derived functor on the category of representations of a crossed product algebra. We investigate the relationship of our theory with Hopf cyclic cohomology and also prove Morita invariance of the Hopf--Hochschild (co)homology.Comment: 19 Pages. Theorem 3.7 and Corollary 4.4 modifie

Global Dimensions of Some Artinian Algebras

In this article we obtain lower and upper bounds for global dimensions of a class of artinian algebras in terms of global dimensions of a finite subset of their artinian subalgebras. Finding these bounds for the global dimension of an artinian algebra $A$ is realized via an explicit algorithm we develop. This algorithm is based on a directed graph (not the Auslander-Reiten quiver) we construct, and it allows us to decide whether an artinian algebra has finite global dimension in good number of cases

Loday--Quillen--Tsygan Theorem for Coalgebras

In this paper we prove that Loday--Quillen--Tsygan Theorem generalizes to the case of coalgebras. Specifically, we show that the Chevalley--Eilenberg--Lie homology of the Lie coalgebra of infinite matrices over a coassociative coalgebra $C$ is generated by the cyclic homology of the underlying coalgebra $C$ as an exterior algebra.Comment: LaTeX, 22 page

The asymptotic Connes-Moscovici characteristic map and the index cocycles

We show that the (even and odd) index cocycles for theta-summable Fredholm modules are in the image of the Connes-Moscovici characteristic map. To show this, we first define a new range of asymptotic cohomologies, and then we extend the Connes-Moscovici characteristic map to our setting. The ordinary periodic cyclic cohomology and the entire cyclic cohomology appear as two instances of this setup. We then construct an asymptotic characteristic class, defined independently from the underlying Fredholm module. Paired with the $K$-theory, the image of this class under the characteristic map yields a non-zero scalar multiple of the index in the even case, and the spectral flow in the odd case

Products in Hopf-Cyclic Cohomology

We construct several pairings in Hopf-cyclic cohomology of (co)module (co)algebras with arbitrary coefficients. The key ideas instrumental in constructing these pairings are the derived functor interpretation of Hopf-cyclic and equivariant cyclic (co)homology, and the Yoneda interpretation of Ext-groups. As a special case of one of these pairings, we recover the Connes-Moscovici characteristic map in Hopf-cyclic cohomology. We also prove that this particular pairing, along with few others, would stay the same if we replace the derived category of (co)cyclic modules with the homotopy category of (special) towers of $X$-complexes, or the derived category of mixed complexes.Comment: 15 page

Bialgebra Cyclic Homology with Coefficients, Part II

This is the second part of the article [math.KT/0408094]. In the first paper, we used the underlying coalgebra structure to develop a cyclic theory. In this paper we define a dual theory by using the algebra structure. We define a cyclic homology theory for triples $(X,B,Y)$ where $B$ is a bialgebra, $X$ is a $B$--comodule algebra and $Y$ is just a stable $B$--module/comodule. We recover the main result of [math.KT/0310088] that these homology theories are dual to each other in the appropriate sense when the bialgebra is a Hopf algebra and the stable coefficient module satisfies anti-Yetter-Drinfeld condition. We also compute this particular homology for the quantum deformation of an arbitrary semi-simple Lie algebra and the Hopf algebra of foliations of codimension $N$ with stable but non-anti-Yetter-Drinfeld coefficients.Comment: 19 pages, LaTeX, no figure

Hopf-cyclic Cohomology of Quantum Enveloping Algebras

In this paper we calculate both the periodic and non-periodic Hopf-cyclic cohomology of Drinfeld-Jimbo quantum enveloping algebra $U_q(\mathfrak{g})$ for an arbitrary semi-simple Lie algebra $\mathfrak{g}$ with coefficients in a modular pair in involution. We show that its Hochschild cohomology is concentrated in a single degree determined by the rank of the Lie algebra $\mathfrak{g}$

Quantum projective space from Toeplitz cubes

From N-tensor powers of the Toeplitz algebra, we construct a multipullback C*-algebra that is a noncommutative deformation of the complex projective space CP(N). Using Birkhoff's Representation Theorem, we prove that the lattice of kernels of the canonical projections on components of the multipullback C*-algebra is free. This shows that our deformation preserves the freeness of the lattice of subsets generated by the affine covering of the complex projective space.Comment: 16 page