A Differential Calculus on the Z3-graded Quantum Group GL q (2)

2015 ◽  
Vol 26 (1) ◽  
pp. 81-96
Author(s):  
Salih Celik ◽  
Fatma Bulut
Keyword(s):  
Quantum Group ◽  
Differential Calculus

scholarly journals Poisson Principal Bundles

2021 ◽  
Author(s):  
Shahn Majid ◽  
◽  
Liam Williams ◽  
Keyword(s):  
Quantum Group ◽  
Lie Group ◽  
Poisson Structure ◽  
Differential Calculus ◽  
Principal Bundle ◽  
Poisson Manifold ◽  
Spin Connection ◽  
Poisson Geometry ◽  
Hopf Fibration ◽  
Principal Bundles

We semiclassicalise the theory of quantum group principal bundles to the level of Poisson geometry. The total space X is a Poisson manifold with Poisson-compatible contravariant connection, the fibre is a Poisson-Lie group in the sense of Drinfeld with bicovariant Poisson-compatible contravariant connection, and the base has an inherited Poisson structure and Poisson-compatible contravariant connection. The latter are known to be the semiclassical data for a quantum differential calculus. The theory is illustrated by the Poisson level of the q-Hopf fibration on the standard q-sphere. We also construct the Poisson level of the spin connection on a principal bundle.

scholarly journals DIFFERENTIAL CALCULUS ON A THREE-PARAMETER OSCILLATOR ALGEBRA

1996 ◽  
Vol 08 (08) ◽  
pp. 1083-1090 ◽  
Author(s):  
MICHÈLE IRAC-ASTAUD
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Differential Calculus ◽  
Heisenberg Algebra ◽  
Invariance Group ◽  
Oscillator Algebra ◽  
Special Case

Two differential calculi are developed on an algebra generalizing the usual q-oscillator algebra and involving three generators and three parameters. They are shown to be invariant under the same quantum group that is extended to a ten-generator Hopf algebra. We discuss the special case where it reduces to a deformation of the invariance group of the Weyl-Heisenberg algebra for which we prove the existence of a constraint between the values of the parameters.

scholarly journals AN INTRODUCTION TO NONCOMMUTATIVE DIFFERENTIAL GEOMETRY ON QUANTUM GROUPS

1993 ◽  
Vol 08 (10) ◽  
pp. 1667-1706 ◽  
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.

scholarly journals DIFFERENTIAL CALCULUS AND CONNECTIONS ON A QUANTUM PLANE AT A CUBIC ROOT OF UNITY

2000 ◽  
Vol 12 (02) ◽  
pp. 227-285 ◽  
Author(s):  
R. COQUEREAUX ◽  
A. O. GARCÍA ◽  
R. TRINCHERO
Keyword(s):  
Representation Theory ◽  
Quantum Group ◽  
Lorentz Group ◽  
Differential Calculus ◽  
Differential Algebra ◽  
Space Time ◽  
Quantum Plane ◽  
Root Of Unity ◽  
Finite Dimensional

We consider the algebra of N×N matrices as a reduced quantum plane on which a finite-dimensional quantum group ℋ acts. This quantum group is a quotient of [Formula: see text], q being an Nth root of unity. Most of the time we shall take N=3; in that case dim(ℋ)=27. We recall the properties of this action and introduce a differential calculus for this algebra: it is a quotient of the Wess–Zumino complex. The quantum group ℋ also acts on the corresponding differential algebra and we study its decomposition in terms of the representation theory of ℋ. We also investigate the properties of connections, in the sense of non commutative geometry, that are taken as 1-forms belonging to this differential algebra. By tensoring this differential calculus with usual forms over space-time, one can construct generalized connections with covariance properties with respect to the usual Lorentz group and with respect to a finite-dimensional quantum group.

scholarly journals INHOMOGENEOUS QUANTUM GROUPS IGLq,r(N): UNIVERSAL ENVELOPING ALGEBRA AND DIFFERENTIAL CALCULUS

1996 ◽  
Vol 11 (06) ◽  
pp. 1019-1056 ◽  
Author(s):  
PAOLO ASCHIERI ◽  
LEONARDO CASTELLANI
Keyword(s):  
Quantum Group ◽  
Differential Calculus ◽  
Matrix Elements ◽  
Dual Algebra ◽  
Enveloping Algebra ◽  
Bicovariant Differential Calculus ◽  
Group Matrix ◽  
Single Structure ◽  
Real Forms ◽  
Inhomogeneous Quantum

A review of the multiparametric linear quantum group GL q,r(N), its real forms, its dual algebra U [ gl q,r(N)] and its bicovariant differential calculus is given in the first part of the paper. We then construct the (multiparametric) linear inhomogeneous quantum group IGL q,r(N) as a projection from GL q,r(N+1) or, equivalently, as a quotient of GL q,r(N+1) with respect to a suitable Hopf algebra ideal. A bicovariant differential calculus on IGL q,r(N) is explicitly obtained as a projection from that on GL q,r(N+1). Our procedure unifies in a single structure the quantum plane coordinates and the q group matrix elements [Formula: see text], and allows one to deduce without effort the differential calculus on the q plane IGL q,r(N)/ GL q,r(N). The general theory is illustrated on the example of IGL q,r(2).

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