A Short Survey of Cyclic Cohomology

This is a short survey of some aspects of Alain Connes' contributions to cyclic cohomology theory in the course of his work on noncommutative geometry over the past 30 years.Comment: To appear in: CMI Proceedings of Noncommutative Geometry Conference in honor of ALAIN CONNES' 60th birthda

Cyclic Homology of DG Coalgebras and a Kuenneth Formula

In this note we extend the cyclic homology functor, and in particular the periodic cyclic homology, to the category of DG (= differential graded) coalgebras. We are partly motivated by the question of products and coproducts in the cyclic homology of algebras, pioneered by A.Connes. As an application we show how one can start from the classical shuffle map in homological algebra and algebraic topology, interpreted by Husemoller, Moore and Stasheff as a morphism of DG coalgebras, and build a theory of products and coproducts in periodic and negative cyclic homology. One of the key ingredients is the idea of X-complex due to Cuntz and Quillen, suitably extended to DG colagebras

On Cyclic Homology of A_{\infty} Algebras

We present an approach to cyclic homology of A_{\infty} algebras. Our main technical tool is the concept of X-complex due to Cuntz and Quillen. This, in particular, enables us to compute the periodic cyclic homology of an A_{\infty} algebra in terms of the periodic cyclic homology of its homology.Comment: 16 page

On Logarithmic Sobolev Inequality for the Noncommutative Two Torus

An analogue of Gross' logarithmic Sobolev inequality for a class of elements of noncommutative two tori is proved

Hopf cyclic cohomology in braided monoidal categories

We extend the formalism of Hopf cyclic cohomology to the context of braided categories. For a Hopf algebra in a braided monoidal abelian category we introduce the notion of stable anti-Yetter-Drinfeld module. We associate a para-cocyclic and a cocyclic object to a braided Hopf algebra endowed with a braided modular pair in involution in the sense of Connes and Moscovici. When the braiding is symmetric the full formalism of Hopf cyclic cohomology with coefficients can be extended to our categorical setting.Comment: 50 pages. One reference added. Proofs are visualized through braiding diagrams. Final version to appear in `Homology, Homotopy and Applications

The Algebra of Formal Twisted Pseudodifferential Symbols and a Noncommutative Residue

We extend the Adler-Manin trace on the algebra of pseudodifferential symbols to a twisted setting.Comment: 29 pages, one reference added, Section 6 is improved and generalize

Noncommutative complex geometry of the quantum projective space

We define holomorphic structures on canonical line bundles of the quantum projective space \qp^{\ell}_q and identify their space of holomorphic sections. This determines the quantum homogeneous coordinate ring of the quantum projective space. We show that the fundamental class of \qp^{\ell}_q is naturally presented by a twisted positive Hochschild cocycle. Finally, we verify the main statements of Riemann-Roch formula and Serre duality for \qp^{1}_q and \qp^{2}_q

Para-Hopf algebroids and their cyclic cohomology

We introduce the concept of {\it para-Hopf algebroid} and define their cyclic cohomology in the spirit of Connes-Moscovici cyclic cohomology for Hopf algebras. Para-Hopf algebroids are closely related to, but different from, Hopf algebroids. Their definition is motivated by attempting to define a cyclic cohomology theory for Hopf algebroids in general. We show that many of Hopf algebraic structures, including the Connes-Moscovici algebra $\mathcal{H}_{FM}$, are para-Hopf algebroids.Comment: Final version to appear in Letters in Mathematical Physics. The name of the article has been changed. Our new class of bialgebroids are now called ``para-Hopf algebroids''. One new example of bialgebroids, due to Connes and Moscovici, are shown to satisfy our axioms of para-Hopf algebroid

On Certain Spectral Invariants of Dirac Operators on Noncommutative Tori

The spectral eta function for certain families of Dirac operators on noncommutative $3$-torus is considered and the regularity at zero is proved. By using variational techniques, we show that $\eta_{D}(0)$ is a conformal invariant. By studying the Laurent expansion at zero of $\text{TR} (|D|^{-z})$, the conformal invariance of $\zeta'_{|D|}(0)$ for noncommutative $3$-torus is proved. Finally, for the coupled Dirac operator, a local formula for the variation $\partial_A\eta_{D+A}(0)$ is derived which is the analogue of the so called induced Chern-Simons term in quantum field theory literature.Comment: 30 page

A Riemann-Roch theorem for the noncommutative two torus

We prove the analogue of the Riemann-Roch formula for the noncommutative two torus $A_{\theta} = C(\mathbb{T}_{\theta}^2)$ equipped with an arbitrary translation invariant complex structure and a Weyl factor represented by a positive element $k\in C^{\infty}(\mathbb{T}_{\theta}^2)$. We consider a topologically trivial line bundle equipped with a general holomorphic structure and the corresponding twisted Dolbeault Laplacians. We define an spectral triple ($A_{\theta}, \mathcal{H}, D)$ that encodes the twisted Dolbeault complex of $A_{\theta}$ and whose index gives the left hand side of the Riemann-Roch formula. Using Connes' pseudodifferential calculus and heat equation techniques, we explicitly compute the $b_2$ terms of the asymptotic expansion of $\text{Tr} (e^{-tD^2})$. We find that the curvature term on the right hand side of the Riemann-Roch formula coincides with the scalar curvature of the noncommutative torus recently defined and computed in \cite{CM1} and \cite{FK2}.Comment: 15 page