We define a new algebra of noncommutative differential forms for any Hopf algebra with an invertible antipode. We prove that there is a one to one correspondence between anti-Yetter-Drinfeld modules, which serve as coefficients for the Hopf cyclic (co)homology, and modules which admit a flat connection with respect to our differential calculus. Thus we show that these coefficient modules can be regarded as ``flat bundles'' in the sense of Connes' noncommutative differential geometry.

2005-12-07math.KTmath.QA

Higher invariants in noncommutative geometry

We give a survey on higher invariants in noncommutative geometry and their applications to differential geometry and topology.

2019-05-29math.KTmath.OA

$\mathbf Z_2$-graded Cech Cohomology in Noncommutative Geometry

The Z_2-graded Cech cohomology theory is considered in the framework of noncommutative geometry over complex number field and in particular the homotopy invariance and Morita invariance are proven.

2004-09-24math.AGmath.KT

The noncommutative geometry of the discrete Heisenberg group

Motivated by the search for new examples of ``noncommutative manifolds'', we study the noncommutative geometry (in the sense of Connes) of the group C*-algebra of the three dimensional discrete Heisenberg group. We present a unified treatment of the K-homology, cyclic cohomology and derivations of this algebra.

2002-01-10math.KTmath.OA

Noncommutative geometry through monoidal categories I

After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasi-coherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, Hopf-Galois extensions) can be given geometric meaning extending their geometric interpretations in the commutative case. On the other hand, we show that some constructions in commutative geometry (e.g. faithfully flat descent theory, principal fibrations, equivariant and infinitesimal geometry) can be interpreted as noncommutative geometric constructions applied to commut…

2007-07-16math.AGmath.KT

The Noncommutative Geometry Generalization of Fundamental Group

A notion of fundamental group of spectral triples has been introduced. The notion uses a noncommutative analogue of unramified coverings. It was shown that in commutative case this fundamental group is a profinite completion of fundamental group of corresponding Riemann manifold.

2006-04-24math.ATmath.KT

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On Type II noncommutative geometry and the JLO character

The Jaffe-Lesniewski-Osterwalder (JLO) character is a homomorphism from K-homology to entire cyclic cohomology. This paper extends the domain of the JLO character to include Type II noncommutative geometry, the geometry represented by unbounded $θ$-summable Breuer-Fredholm modules; and shows that the JLO character coincides with the Chern-Connes character as a class in entire cyclic cohomolgoy.

2010-10-25math-phmath.KTmath.MP

Riemannian manifolds in noncommutative geometry

We present a definition of Riemannian manifold in noncommutative geometry. Using products of unbounded Kasparov modules, we show one can obtain such Riemannian manifolds from noncommutative spin^c manifolds; and conversely, in the presence of a spin^c structure. We also show how to obtain an analogue of Kasparov's fundamental class for a Riemannian manifold, and the associated notion of Poincaré duality. Along the way we clarify the bimodule and first-order conditions for spectral triples.

2012-03-08math.DGmath.KTmath.OA