Bijectivity of the canonical map for the noncommutative instanton bundle

It is shown that the quantum instanton bundle introduced in Commun. Math. Phys. 226, 419-432 (2002) has a bijective canonical map and is, therefore, a coalgebra Galois extension.Comment: Latex, 12 pages. Published versio

Star product formula of theta functions

As a noncommutative generalization of the addition formula of theta functions, we construct a class of theta functions which are closed with respect to the Moyal star product of a fixed noncommutative parameter. These theta functions can be regarded as bases of the space of holomorphic homomorphisms between holomorphic line bundles over noncommutative complex tori.Comment: 12 page

The Hopf algebra structure of the Z3_3-graded quantum supergroup GLq,j(1∣1)_{q,j}(1|1)

In this work, we give some features of the Z$_3$-graded quantum supergroup

The 3D Spin Geometry of the Quantum Two-Sphere

We study a three-dimensional differential calculus on the standard Podles quantum two-sphere S^2_q, coming from the Woronowicz 4D+ differential calculus on the quantum group SU_q(2). We use a frame bundle approach to give an explicit description of the space of forms on S^2_q and its associated spin geometry in terms of a natural spectral triple over S^2_q. We equip this spectral triple with a real structure for which the commutant property and the first order condition are satisfied up to infinitesimals of arbitrary order.Comment: v2: 25 pages; minor change

h-deformation of GL(1|1)

h-deformation of (graded) Hopf algebra of functions on supergroup GL(1|1) is introduced via a contration of GL_q (1|1). The deformation parameter h is odd (grassmann). Related differential calculus on h-superplane is presented.Comment: latex file, 8 pages, minor change

Non-commutative connections of the second kind

A connection-like objects, termed {\em hom-connections} are defined in the realm of non-commutative geometry. The definition is based on the use of homomorphisms rather than tensor products. It is shown that hom-connections arise naturally from (strong) connections in non-commutative principal bundles. The induction procedure of hom-connections via a map of differential graded algebras or a differentiable bimodule is described. The curvature for a hom-connection is defined, and it is shown that flat hom-connections give rise to a chain complex.Comment: 13 pages, LaTe

The Noncommutative Geometry of the Quantum Projective Plane

We study the spectral geometry of the quantum projective plane CP^2_q, a deformation of the complex projective plane CP^2, the simplest example of a spin^c manifold which is not spin. In particular, we construct a Dirac operator D which gives a 0^+ summable spectral triple, equivariant under U_q(su(3)). The square of D is a central element for which left and right actions on spinors coincide, a fact that is exploited to compute explicitly its spectrum.Comment: v2: 26 pages. Paper completely reorganized; no major change, several minor one

On Spinors Transformations

We begin showing that for even dimensional vector spaces $V$ all automorphisms of their Clifford algebras are inner. So all orthogonal transformations of $V$ are restrictions to $V$ of inner automorphisms of the algebra. Thus under orthogonal transformations $P$ and $T$ - space and time reversal - all algebra elements, including vectors $v$ and spinors $\varphi$, transform as $v \to x v x^{-1}$ and $\varphi \to x \varphi x^{-1}$ for some algebra element $x$. We show that while under combined $PT$ spinor $\varphi \to x \varphi x^{-1}$ remain in its spinor space, under $P$ or $T$ separately $\varphi$ goes to a 'different' spinor space and may have opposite chirality. We conclude with a preliminary characterization of inner automorphisms with respect to their property to change, or not, spinor spaces.Comment: Minor changes to propositions 1 and

Twisted Hochschild Homology of Quantum Hyperplanes

We calculate the Hochschild dimension of quantum hyperplanes using the twisted Hochschild homology.Comment: 12 pages, LaTe