Differential calculi on noncommutative bundles

Letters in Mathematical Physics, 2006

1995

Differential calculi are obtained for quantum homogeneous spaces by extend- ing Woronowicz' approach to the present context. Representation theoretical properties of the differential calculi are investigated. Connections on quantum homogeneous vector bundles are classified and explicitly constructed by using the theory of projective modules.

arXiv (Cornell University), 2022

In this paper, we revise the concept of noncommutative vector fields introduced previously in [1, 2], extending the framework, adding new results and clarifying the old ones. Using appropriate algebraic tools certain shortcomings in the previous considerations are filled and made more precise. We focus on the correspondence between so-called Cartan pairs and first-order differentials. The case of free bimodules admitting more friendly "coordinate description" and their braiding is considered in more detail. Bimodules of right/left universal vector fields are explicitly constructed.

Geometric and Topological Methods for Quantum Field Theory, 2003

2018

Noncommutative geometry is the idea that when geometry is done in terms of coordinate algebras, one does not really need the algebra to be commutative. We provide an introduction to the relevant mathematics from a constructive ‘differential algebra’ point of view that works over general fields and includes the noncommutative geometry of quantum groups as well as of finite groups. We also mention applications to models of quantum spacetime.

This paper deals with sheaves of differential operators on noncommutative algebras. The sheaves are defined by quotienting a the tensor algebra of vector fields (suitably deformed by a covariant derivative) to ensure zero curvature. As an example we can obtain enveloping algebra like relations for Hopf algebras with differential structures which are not bicovariant. Symbols of differential operators are defined, but not studied. These sheaves are shown to be in the center of as category of bimodules with flat bimodule covariant derivatives. Also holomorphic differential operators are considered, though without the quotient to ensure zero curvature.

Classical and Quantum Gravity, 1995

A construction is proposed for linear connections on non-commutative algebras. The construction relies on a generalisation of the Leibnitz rules of commutative geometry and uses the bimodule structure of Ω 1 . A special role is played by the extension to the framework of non-commutative geometry of the permutation of two copies of Ω 1 . The construction of the linear connection as well as the definition of torsion and curvature is first proposed in the setting of the derivations based differential calculus of Dubois-Violette and then a generalisation to the framework proposed by Connes as well as other non-commutative differential calculi is suggested. The covariant derivative obtained admits an extension to the tensor product of several copies of Ω 1 . These constructions are illustrated with the example of the algebra of n × n matrices.