# Lectures on differentials, generalized differentials and on some examples related to theoretical physics

@article{DuboisViolette2000LecturesOD, title={Lectures on differentials, generalized differentials and on some examples related to theoretical physics}, author={Michel Dubois-Violette}, journal={arXiv: Quantum Algebra}, year={2000} }

These notes contain a survey of some aspects of the theory of differential modules and complexes as well as of their generalization, that is, the theory of $N$-differential modules and $N$-complexes. Several applications and examples coming from physics are discussed. The commun feature of these physical applications is that they deal with the theory of constrained or gauge systems. In particular different aspects of the BRS methods are explained and a detailed account of the $N$-complexes…

## 39 Citations

N-Complex, Graded q-Differential Algebra and N-Connection on Modules

- Mathematics
- 2015

It is well known that given a differential module E with a differential d we can measure the non-exactness of this differential module by its homologies which are based on the key relation d2=0. This…

Generalization of connection based on the concept of graded q-differential algebra

- Mathematics
- 2010

We propose a generalization of the concept of connection form by means of a graded q-differential algebra Wq, where q is a primitive Nth root of unity, and develop the concept of curvature N-form for…

Generalization of superconnection in noncommutative geometry

- MathematicsProceedings of the Estonian Academy of Sciences. Physics. Mathematics
- 2006

We propose the notion of a ZN -connection, where N � 2, which can be viewed as a generalization of the notion of a Z2-connection or superconnection. We use the algebraic approach to the theory of…

QUANTUM GROUPS AND DIFFERENTIAL FORMS

- Mathematics
- 2005

ABSTRACT We construct a quantum semigroup and an algebra of forms appropriate for the generalised homological algebra of N-complexes (Kapranov, arXiv:q-alg/9611005). This is an analogue to the…

Geometric approach to BRST-symmetry and ℤN-generalization of superconnection

- Mathematics
- 2006

Abstract We propose a geometric approach to the BRST-symmetries of the Lagrangian of a topological quantum field theory on a four dimensional manifold based on the formalism of superconnections.…

Non-associative gauge theory and higher spin interactions

- Mathematics
- 2005

We give a framework to describe gauge theory on a certain class of commutative but non-associative fuzzy spaces. Our description is in terms of an abelian gauge connection valued in the algebra of…

2 Submanifolds and Quotient Manifolds in Noncommutative Differential Calculus : Definitions and a Brief Review

- Mathematics
- 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…

Generalized Forms, Connections, and Gauge Theories

- Mathematics
- 2003

Generalized differential forms of type N = 2, and flat generalized connections are used to describe the SO(p, q) form of Cartan's structure equations for metric geometries, source-free Yang-Mills…

Noncommutative geometry of Poisson structures

- Mathematics
- 2007

Geometric, algebraic, and homological properties of Poisson structures on smooth manifolds are studied. Noncommutative (NC) foundations of these structures are introduced for associative Poisson…

Noncommutative Symplectic Foliation, Bott Connection and Phase Space Reduction

- Mathematics
- 2004

We investigate the geometric, algebraic and homological properties of Poisson structures on smooth manifolds and introduce noncommutative foundations of these structures for associative Poisson…

## References

SHOWING 1-10 OF 78 REFERENCES

On the q-analog of homological algebra

- Mathematics
- 1996

This is an attempt to generalize some basic facts of homological algebra to the case of "complexes" in which the differential satisfies the condition $d^N=0$ instead of the usual $d^2=0$. Instead of…

Tensor Fields of Mixed Young Symmetry Type¶and N-Complexes

- Mathematics
- 2002

Abstract: We construct N-complexes of non-completely antisymmetric irreducible tensor fields on ℝD which generalize the usual complex (N=2) of differential forms. Although, for N≥ 3, the generalized…

Generalized Cohomology for Irreducible Tensor Fields of Mixed Young Symmetry Type

- Mathematics
- 1999

We construct N-complexes of noncompletely antisymmetric irreducible tensor fields on ℝD, thereby generalizing the usual complex (N=2) of differential forms. These complexes arise naturally in the…

Lectures on graded differential algebras and noncommutative geometry

- Mathematics
- 1999

These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also…

Some remarks on BRS transformations, anomalies and the cohomology of the Lie algebra of the group of gauge transformations

- Mathematics
- 1983

We show that ghosts in gauge theories can be interpreted as Maurer-Cartan forms in the infinite dimensional group ℊ of gauge transformations. We examine the cohomology of the Lie algebra of ℊ and…

Generalized differential spaces withdN=0 and theq-differential calculus

- Mathematics
- 1996

We present some results concerning the generalized homologies associated with nilpotent endomorphismsd such thatdN=0 for some integerN≥2. We then introduce the notion of gradedq-differential algebra…

Homologie cyclique et K-théorie

- Mathematics
- 1987

This book is an expanded version of some ideas related to the general problem of characteristic classes in the framework of Chern-Weil theory. These ideas took their origins independtly from the work…

Quantization of Gauge Systems

- Physics
- 1992

This is a systematic study of the classical and quantum theories of gauge systems. It starts with Dirac's analysis showing that gauge theories are constrained Hamiltonian systems. The classical…

Homological perturbation theory and the algebraic structure of the antifield-antibracket formalism for gauge theories

- Mathematics
- 1990

The algebraic structure of the antifield-antibracket formalism for both reducible and irreducible gauge theories is clarified. This is done by using the methods of Homological Perturbation Theory…