Differential operators and differential calculus in quantum groups

Improving on an earlier proposal, we construct the gauge theories of the quantum groups U q(N). We find that these theories are also consistent with an ordinary (commuting) space-time. The bicovariance conditions of the quantum differential calculus are essential in our construction. The gauge potentials and the field strengths are q-commuting "fields," and satisfy q-commutation relations with the gauge parameters. The transformation rules of the potentials generalize the ordinary infinitesimal gauge variations. For particular deformations of U (N) ("minimal deformations"), the algebra of quantum gauge variations is shown to close, provided the gauge parameters satisfy appropriate q-commutations. The q-Lagrangian invariant under the U q(N) variations has the Yang–Mills form [Formula: see text], the "quantum metric" gij being a generalization of the Killing metric.

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Manin Triples and Differential Operators on Quantum Groups

Tokyo Journal of Mathematics ◽

10.3836/tjm/1374497512 ◽

2013 ◽

Vol 36 (1) ◽

pp. 49-83 ◽

Cited By ~ 1

Author(s):

Toshiyuki TANISAKI

Keyword(s):

Quantum Groups ◽

Differential Operators ◽

Manin Triples

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R-matrix approach to differential calculus on quantum groups

Physics of Particles and Nuclei ◽

10.1134/1.953040 ◽

1997 ◽

Vol 28 (3) ◽

pp. 267 ◽

Cited By ~ 6

Author(s):

A. P. Isaev

Keyword(s):

Quantum Groups ◽

Differential Calculus ◽

Matrix Approach ◽

R Matrix

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AN INTRODUCTION TO NONCOMMUTATIVE DIFFERENTIAL GEOMETRY ON QUANTUM GROUPS

International Journal of Modern Physics A ◽

10.1142/s0217751x93000692 ◽

1993 ◽

Vol 08 (10) ◽

pp. 1667-1706 ◽

Cited By ~ 65

Author(s):

PAOLO ASCHIERI ◽

LEONARDO CASTELLANI

Keyword(s):

Differential Geometry ◽

Quantum Group ◽

Explicit Form ◽

Quantum Groups ◽

Differential Calculus ◽

Gauge Theories ◽

Classical Case ◽

Contraction Operator ◽

Lie Derivative ◽

Tensor Fields

We give a pedagogical introduction to the differential calculus on quantum groups by stressing at all stages the connection with the classical case (q→1 limit). The Lie derivative and the contraction operator on forms and tensor fields are found. A new, explicit form of the Cartan-Maurer equations is presented. The example of a bicovariant differential calculus on the quantum group GL q(2) is given in detail. The softening of a quantum group is considered, and we introduce q curvatures satisfying q Bianchi identities, a basic ingredient for the construction of q gravity and q gauge theories.

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Differential Calculus onN-Graded Manifolds

Journal of Mathematics ◽

10.1155/2017/8271562 ◽

2017 ◽

Vol 2017 ◽

pp. 1-19

Author(s):

G. Sardanashvily ◽

W. Wachowski

Keyword(s):

Differential Calculus ◽

Differential Operators ◽

Differential Forms ◽

Commutative Rings ◽

Rham Complex ◽

Grassmann Algebras ◽

De Rham ◽

Linear Differential ◽

Graded Manifold ◽

Graded Manifolds

The differential calculus, including formalism of linear differential operators and the Chevalley–Eilenberg differential calculus, overN-graded commutative rings and onN-graded manifolds is developed. This is a straightforward generalization of the conventional differential calculus over commutative rings and also is the case of the differential calculus over Grassmann algebras and onZ2-graded manifolds. We follow the notion of anN-graded manifold as a local-ringed space whose body is a smooth manifoldZ. A key point is that the graded derivation module of the structure ring of graded functions on anN-graded manifold is the structure ring of global sections of a certain smooth vector bundle over its bodyZ. Accordingly, the Chevalley–Eilenberg differential calculus on anN-graded manifold provides it with the de Rham complex of graded differential forms. This fact enables us to extend the differential calculus onN-graded manifolds to formalism of nonlinear differential operators, by analogy with that on smooth manifolds, in terms of graded jet manifolds ofN-graded bundles.

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Recent developments on Pseudo-Differential Operators (I)

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.46.2015.1707 ◽

2015 ◽

Vol 46 (1) ◽

pp. 1-30 ◽

Cited By ~ 1

Author(s):

D.-C. Chang ◽

W. RUNGROTTHEERA ◽

B.-W. SCHULZE

Keyword(s):

Boundary Value Problems ◽

Differential Calculus ◽

Differential Operators ◽

Elliptic Boundary ◽

Higher Order ◽

Symbolic Calculus ◽

Elliptic Boundary Value ◽

The Past ◽

Pseudo Differential Operators ◽

Recent Developments

In recent years the analysis of (pseudo-)differential operators on manifolds with second and higher order corners made considerable progress, and essential new structures have been developed. The main objective of this series of paper is to give a survey on the development of this theory in the past twenty years. We start with a brief background of the theory of pseudo-differential operators which including its symbolic calculus on $\R^n$. Next we introduce pseudo-differential calculus with operator-valued symbols. This allows us to discuss elliptic boundary value problems on smooth domains in $\R^n$ and elliptic problems on manifolds. This paper is based on the first part of lectures given by the authors while they visited the National Center for Theoretical Sciences in Hsinchu, Taiwan during May-July of 2014.

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A new approach to the r-Whitney numbers by using combinatorial differential calculus

Acta Universitatis Sapientiae Mathematica ◽

10.2478/ausm-2019-0029 ◽

2019 ◽

Vol 11 (2) ◽

pp. 387-418

Author(s):

Miguel A. Méndez ◽

José L. Ramírez

Keyword(s):

Differential Calculus ◽

Differential Operators ◽

Family Study ◽

Stirling Numbers ◽

Umbral Calculus ◽

Combinatorial Interpretation ◽

New Approach ◽

Whitney Numbers ◽

Combinatorial Interpretations ◽

Binomial Type

Abstract In the present article we introduce two new combinatorial interpretations of the r-Whitney numbers of the second kind obtained from the combinatorics of the differential operators associated to the grammar G := {y → yxm, x → x}. By specializing m = 1 we obtain also a new combinatorial interpretation of the r-Stirling numbers of the second kind. Again, by specializing to the case r = 0 we introduce a new generalization of the Stirling number of the second kind and through them a binomial type family of polynomials that generalizes Touchard’s polynomials. Moreover, we recover several known identities involving the r-Dowling polynomials and the r-Whitney numbers using the combinatorial differential calculus. We construct a family of posets that generalize the classical Dowling lattices. The r-Withney numbers of the first kind are obtained as the sum of the Möbius function over elements of a given rank. Finally, we prove that the r-Dowling polynomials are a Sheffer family relative to the generalized Touchard binomial family, study their umbral inverses, and introduce [m]-Stirling numbers of the first kind. From the relation between umbral calculus and the Riordan matrices we give several new combinatorial identities

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