In this paper we study the cyclic cohomology of certain ×-Hopf algebras: universal enveloping algebras, quantum algebraic tori, the Connes-Moscovici ×-Hopf algebroids and the Kadison bialgebroids. Introducing their stable anti Yetter-Drinfeld modules and cocyclic modules, we compute their cyclic cohomology. Furthermore, we provide a pairing for the cyclic cohomology of ×-Hopf algebras which generalizes the Connes-Moscovici characteristic map to ×-Hopf algebras. This enables us to transfer the ×-Hopf algebra cyclic cocycles to algebra cyclic cocycles.

Algebras associated with quantum electrodynamics and other gauge theories share some mathematical features with T-duality. Exploiting this different perspective and some category theory, the full algebra of fermions and bosons can be regarded as a braided Clifford algebra over a braided commutative boson algebra, sharing much of the structure of ordinary Clifford algebras.

We construct explicit generators of the K-theory and K-homology of the coordinate algebras of ‘functions’ on the quantum projective spaces. We also sketch a construction of unbounded Fredholm modules, that is to say Diraclike operators and spectral triples of any positive real dimension.

We review basic notions and methods of noncommutative geometry and their applications to analysis and geometry on foliated manifolds.

We construct invariants of relative K-theory classes of multiparameter dependent pseudodifferential operators, which recover and generalize Melrose's divisor flow and its higher odd-dimensional versions of Lesch and Pflaum. These higher divisor flows are obtained by means of pairing the relative K-theory modulo the symbols with the cyclic cohomological characters of relative cycles constructed out of the regularized operator trace together with its symbolic boundary. Besides giving a clear and conceptual explanation to the essential features of the divisor flows, namely homotopy invariance, additivity and integrality, this construction allows to uncover the previously unknown even-dimensional counterparts. Furthermore, it confers to the totality of these invariants a purely topological interpretation, that of implementing the classical Bott periodicity isomorphisms in a manner compatible with the suspension isomorphisms in both K-theory and in cyclic cohomology. We also give a precise formulation, in terms of a natural Clifford algebraic suspension, for the relationship between the higher divisor flows and the spectral flow.

In this note we show that the crucial orientation condition forcommutative geometries fails for the natural commutative spectral triple of an orbifold M/G.

To any finite group $\Gamma \subset {SL}_2({\mathbb {C}})$ and each element τ in the center of the group algebra of Γ, we associate a category, ${\mathcal {C}}\! oh({\mathbb {P}}^2_{_{\! \Gamma ^{\! },^{\! }\tau }},{\mathbb {P}}^1).$ It is defined as a suitable quotient of the category of graded modules over (a graded version of) the deformed preprojective algebra introduced by Crawley-Boevey and Holland. The category ${\mathcal {C}}\! oh({\mathbb {P}}^2_{_{\! \Gamma ^{\! },^{\! }\tau }},{\mathbb {P}}^1)$ should be thought of as the category of coherent sheaves on a ‘noncommutative projective space’, ${\mathbb {P}}^2_{_{\! \Gamma ^{\! },^{\! }\tau }},$ equipped with a framing at ${\mathbb {P}}^1$, the line at infinity. Our first result establishes an isomorphism between the moduli space of torsion free objects of ${\mathcal {C}}\! oh({\mathbb {P}}^2_{_{\! \Gamma ^{\! },^{\! }\tau }},{\mathbb {P}}^1)$ and the Nakajima quiver variety arising from Γ via the McKay correspondence. We apply the above isomorphism to deduce a generalization of the Crawley-Boevey and Holland conjecture, saying that the moduli space of `rank 1' projective modules over the deformed preprojective algebra is isomorphic to a particular quiver variety. This reduces, for $\Gamma =\{ 1\}$, to the recently obtained parametrisation of the isomorphism classes of right ideals in the first Weyl algebra, $\sf {A_1}$, by points of the Calogero–Moser space, due to Cannings and Holland and Berest and Wilson. Our approach is algebraic and is based on a monadic description of torsion free sheaves on ${\mathbb {P}}^2_{_{\! \Gamma ^{\! },^{\! }\tau }}$. It is totally different from the one used by Berest and Wilson, involving $\tau $-functions.