Vector fields and differential operators: noncommutative case

We consider differential operators over a noncommutative algebra A generated by vector fields. These are shown to form a unital associative algebra of differential operators, and act on A-modules E with covariant derivative. We use the repeated differentials given in the paper to give a definition of noncommutative Sobolev space for modules with connection and Hermitian inner product. The tensor algebra of vector fields, with a modified bimodule structure and a bimodule connection, is shown to lie in the centre of the bimodule connection category A E A , and in fact to be an algebra in the centre. The crossing natural transformation in the definition of the centre of the category is related to the action of the differential operators on bimodules with connection.

Czechoslovak journal of physics, 2003

The aim of this paper is to avoid some difficulties, related with the Lie bracket, in the definition of vector fields in a noncommutative setting, as they were defined by Woronowicz, Schmüdgen-Schüler and Aschieri-Schupp. We extend the definition of vector fields to ...

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.

1995

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.

International Journal of Mathematics and Mathematical Sciences, 2003

A nonassociative algebra endowed with a Lie bracket, called atorsion algebra, is viewed as an algebraic analog of a manifold with an affine connection. Its elements are interpreted as vector fields and its multiplication is interpreted as a connection. This provides a framework for differential geometry on a formal manifold with a formal connection. A torsion algebra is a natural generalization of pre-Lie algebras which appear as the “torsionless” case. The starting point is the observation that the associator of a nonassociative algebra is essentially the curvature of the corresponding Hochschild quasicomplex. It is a cocycle, and the corresponding equation is interpreted as Bianchi identity. The curvature-associator-monoidal structure relationships are discussed. Conditions on torsion algebras allowing to construct an algebra of functions, whose algebra of derivations is the initial Lie algebra, are considered. The main example of a torsion algebra is provided by the pre-Lie algeb...

2008

We study noncommutative generalizations of such notions of the classical symplectic geometry as degenerate Poisson structure, Poisson submanifold and quotient manifold, symplectic foliation and symplectic leaf, for associative Poisson algebras. We give the full description of the family of Poisson structures on the endomorphism algebra of a vector bundle and study the above structures in the case of this algebra. Introduce the notion of generalized center of Poisson algebra as a subspace of the space of generalized functions (distributions) on a Poisson manifold and study its relation with the geometrical and homological properties of a singular Poisson structure.

Journal of Geometry and Physics, 1989

The structure of amanifold can be encoded in the commutative algebra of functions on the manifold it sell-this is usual-. In the case of a non com.mut.ative algebra thereis no underlying manifold and the usual concepts and tools of diffe.rential geometry (differentialforms, De Rham cohomology, vector bundles, connections, elliptic operators, index theory.. .) have to be generalized. This is the subject of non commutative differential geometry and is believed to be of fundamental importance in our understanding of quantum field theories. The presentpaper is an introduction for the non specialist and a review oftheprincipal results on the field.

Symmetry, 2022

This paper shows how gauge theoretic structures arise in a noncommutative calculus where the derivations are generated by commutators. These patterns include Hamilton’s equations, the structure of the Levi–Civita connection, and generalizations of electromagnetism that are related to gauge theory and with the early work of Hermann Weyl. The territory here explored is self-contained mathematically. It is elementary, algebraic, and subject to possible generalizations that are discussed in the body of the paper.