Lie bracket of vector fields in noncommutative geometry

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.

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.

2006

Journal of Geometry and Physics, 1993

This is an introduction to the old and new concepts of non-commutative (N.C.) geometry. We review the ideas underlying N.C. measure and topology, N.C. differential calculus, N.C. connections on N.C. vector bundles, and N.C. Riemannian geometry by following A. Connes' point of view.

Journal of Pure and Applied Algebra, 1997

In commutative differential geometry the Frölicher-Nijenhuis bracket computes all kinds of curvatures and obstructions to integrability. In [1] the Frölicher-Nijenhuis bracket was developed for universal differential forms of noncommutative algebras, and several applications were given. In this paper this bracket and the Frölicher-Nijenhuis calculus will be developed for several kinds of differential graded algebras based on derivations, which were introduced in [6].

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.

Nonlinear Differential Equations and Applications NoDEA

If the vector fields f1, f2 are locally Lipschitz, the classical Lie bracket [f1, f2] is defined only almost everywhere. However, it has been shown that, by means of a set-valued Lie bracket [f1, f2]set (which is defined everywhere), one can generalize classical results like the Commutativity theorem and Frobenius' theorem, as well as a Chow-Rashevski's theorem involving Lie brackets of degree 2 (we call 'degree' the number of vector fields contained in a formal bracket). As it might be expected, these results are consequences of the validity of an asymptotic formula similar to the one holding true in the regular case. Aiming to more advanced applications-say, a general Chow-Rashevski's theorem or higher order conditions for optimal controls-we address here the problem of defining, for any m > 2 and any formal bracket B of degree m, a Lie bracket B(f1,. .. , fm) corresponding to vector fields (f1,. .. , fm) lacking classical regularity requirements. A major complication consists in finding the right extension of the degree 2 bracket, namely a notion of bracket which admits an asymptotic formula. In fact, it is known that a mere iteration of the construction performed for the case m = 2 is not compatible with the validity of an asymptotic formula. We overcome this difficulty by introducing a set-valued bracket x → Bset(f1,. .. , fm)(x), defined at This article is part of the topical collection "Hyperbolic PDEs, Fluids, Transport and Applications: Dedicated to Alberto Bressan for his 60th birthday" guest edited by

2003

The purpose of this paper is to put into a noncommutative context basic notions related to vector fields from classical differential geometry. The manner of exposition is an attempt to make the material as accessible as possible to classical geometers. The definition of vector field used is a specialisation of the Cartan pair definition, and the paper relies on the

Eprint Arxiv Quant Ph 0305150, 2003

The development of Noncommutative geometry is creating a reworking and new possibilities in physics. This paper identifies some of the commutation and derivation structures that arise in particle and field interactions and fundamental symmetries. The requirements of coexisting structures, and their consistency, produce a mathematical framework that underlies a fundamental physics theory. Among other developments in Quantum theory of particles and fields are the symmetries of gauge fields and the Fermi-Bose symmetry of particles. These involve a gauge covariant derivation and the action functionals ; and commutation algebras and Bogoliubov transforms. The non commutative Theta form introduces an additional and fundamental structure. This paper obtains the interrelations of the various structures; and the conditions for the symmetries of Fermionic/Bosonic particles interacting with Yang-Mills gauge fields. Many example physical systems are being solved , and the mathematical formalism is being created to understand the fundamental basis of physics. 1.Introduction The mathematical structures of the physics of particles and fields were developed using commutative and non commutative algebra, and Euclidean and non Euclidean Geometry. This led to Quantum Mechanics and General Relativity,respectively. The Quantum Field theory of Gauge Fields describes all fundamental interactions, including gravity, as holonomy and action integrals. It has succeeded phenomenologically, inspite of some difficulties. Consistency requirements have led to a number of symmetries, including supersymmetry. Loop space quantum gravity and string and brane theories have evolved as a development of quantum theory of interactions. These are also connected to the evolving subject of non commutative geometry.[Ref ] The dynamical variables in a quantum theory have a commutation algebra. A non commutative structure has been introduced in a wide variety of physics ; with length scales from Planck length in quantum space time, to magnetic length in quantum Hall effect. The new (non)commutation structure introduces a derivation (as a bracket operation), which acts in addition to the Lie and covariant derivatives. In the spacetime manifold , a discrete topology and a length scale parameter cause changes in the definitions of the metric tensor,Riemann tensor, Ricci tensor and the Einstein equations.Will

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...

Zeitschrift f�r Physik C Particles and Fields, 1997

We introduce a category of noncommutative bundles. To establish geometry in this category we construct suitable noncommutative differential calculi on these bundles and study their basic properties. Furthermore we define the notion of a connection with respect to a differential calculus and consider questions of existence and uniqueness. At the end these constructions are applied to basic examples of noncommutative bundles over a coquasitriangular Hopf algebra.

Pacific Journal of Mathematics, 2016

We show that the quantisation of a connected simply-connected Poisson-Lie group admits a left-covariant noncommutative differential structure at lowest deformation order if and only if the dual of its Lie algebra admits a pre-Lie algebra structure. As an example, we find a pre-Lie algebra structure underlying the standard 3D differential structure on Cq[SU 2 ]. At the noncommutative geometry level we show that the enveloping algebra U (m) of a Lie algebra m, viewed as quantisation of m * , admits a connected differential exterior algebra of classical dimension if and only if m admits a pre-Lie algebra. We give an example where m is solvable and we extend the construction to the quantisation of tangent and cotangent spaces of Poisson-Lie groups by using bicross-sum and bosonization of Lie bialgebras.

Classical and Quantum Gravity, 2005

1995

A general definition of a bimodule connection in noncommutative geometry has been recently proposed. For a given algebra this definition is compared with the ordinary definition of a connection on a left module over the associated enveloping algebra. The corresponding curvatures are also compared.

Journal of Mathematical Physics, 2008