14,200 research outputs found
Divergences on projective modules and non-commutative integrals
A method of constructing (finitely generated and projective) right module
structure on a finitely generated projective left module over an algebra is
presented. This leads to a construction of a first order differential calculus
on such a module which admits a hom-connection or a divergence. Properties of
integrals associated to this divergence are studied, in particular the formula
of integration by parts is derived. Specific examples include inner calculi on
a noncommutative algebra, the Berezin integral on the supercircle and integrals
on Hopf algebras.Comment: 13 pages; v2 construction of projective modules has been generalise
Holomorphic structures on the quantum projective line
We show that much of the structure of the 2-sphere as a complex curve
survives the q-deformation and has natural generalizations to the quantum
2-sphere - which, with additional structures, we identify with the quantum
projective line. Notably among these is the identification of a quantum
homogeneous coordinate ring with the coordinate ring of the quantum plane. In
parallel with the fact that positive Hochschild cocycles on the algebra of
smooth functions on a compact oriented 2-dimensional manifold encode the
information for complex structures on the surface, we formulate a notion of
twisted positivity for twisted Hochschild and cyclic cocycles and exhibit an
explicit twisted positive Hochschild cocycle for the complex structure on the
sphere.Comment: 22 pages, no figures. Published in IMR
Quantisation of twistor theory by cocycle twist
We present the main ingredients of twistor theory leading up to and including
the Penrose-Ward transform in a coordinate algebra form which we can then
`quantise' by means of a functorial cocycle twist. The quantum algebras for the
conformal group, twistor space CP^3, compactified Minkowski space CMh and the
twistor correspondence space are obtained along with their canonical quantum
differential calculi, both in a local form and in a global *-algebra
formulation which even in the classical commutative case provides a useful
alternative to the formulation in terms of projective varieties. We outline how
the Penrose-Ward transform then quantises. As an example, we show that the
pull-back of the tautological bundle on CMh pulls back to the basic instanton
on S^4\subset CMh and that this observation quantises to obtain the
Connes-Landi instanton on \theta-deformed S^4 as the pull-back of the
tautological bundle on our \theta-deformed CMh. We likewise quantise the
fibration CP^3--> S^4 and use it to construct the bundle on \theta-deformed
CP^3 that maps over under the transform to the \theta-deformed instanton.Comment: 68 pages 0 figures. Significant revision now has detailed formulae
for classical and quantum CP^
First Order Calculi with Values in Right--Universal Bimodules
The purpose of this note is to show how calculi on unital associative algebra
with universal right bimodule generalize previously studied constructions by
Pusz and Woronowicz [1989] and by Wess and Zumino [1990] and that in this
language results are in a natural context, are easier to describe and handle.
As a by--product we obtained intrinsic, coordinate--free and basis--independent
generalization of the first order noncommutative differential calculi with
partial derivatives.Comment: 13 pages in TeX, the macro package bcp.tex included, to be published
in Banach Center Publication, the Proceedings of Minisemester on Quantum
Groups and Quantum Spaces, November 199
Projective Group Algebras
In this paper we apply a recently proposed algebraic theory of integration to
projective group algebras. These structures have received some attention in
connection with the compactification of the theory on noncommutative tori.
This turns out to be an interesting field of applications, since the space
of the equivalence classes of the vector unitary irreducible
representations of the group under examination becomes, in the projective case,
a prototype of noncommuting spaces. For vector representations the algebraic
integration is equivalent to integrate over . However, its very
definition is related only at the structural properties of the group algebra,
therefore it is well defined also in the projective case, where the space has no classical meaning. This allows a generalization of the usual group
harmonic analysis. A particular attention is given to abelian groups, which are
the relevant ones in the compactification problem, since it is possible, from
the previous results, to establish a simple generalization of the ordinary
calculus to the associated noncommutative spaces.Comment: 24 pages, Late
q-Deformed quaternions and su(2) instantons
We have recently introduced the notion of a q-quaternion bialgebra and shown
its strict link with the SO_q(4)-covariant quantum Euclidean space R_q^4.
Adopting the available differential geometric tools on the latter and the
quaternion language we have formulated and found solutions of the
(anti)selfduality equation [instantons and multi-instantons] of a would-be
deformed su(2) Yang-Mills theory on this quantum space. The solutions depend on
some noncommuting parameters, indicating that the moduli space of a complete
theory should be a noncommutative manifold. We summarize these results and add
an explicit comparison between the two SO_q(4)-covariant differential calculi
on R_q^4 and the two 4-dimensional bicovariant differential calculi on the bi-
(resp. Hopf) algebras M_q(2),GL_q(2),SU_q(2), showing that they essentially
coincide.Comment: Latex file, 18 page
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