Bicovariant differential calculus on quantum groups and wave mechanics

1992 ◽  
Vol 42 (12) ◽  
pp. 1279-1288
Author(s):  
Ursula Carow-Watamura ◽  
Satoshi Watamura ◽  
Arthur Hebecker ◽  
Michael Schlieker ◽  
Wolfgang Weich
Keyword(s):  
Quantum Groups ◽  
Differential Calculus ◽  
Wave Mechanics ◽  
Bicovariant Differential Calculus

scholarly journals The Quantum Cartan Algebra Associated to a Bicovariant Differential Calculus

2011 ◽  
Vol 68 (3) ◽  
pp. 319-346
Author(s):  
Lucio S. Cirio ◽  
Chiara Pagani ◽  
Alessandro Zampini
Keyword(s):  
Differential Calculus ◽  
Bicovariant Differential Calculus

scholarly journals Uq(N) GAUGE THEORIES

1994 ◽  
Vol 09 (30) ◽  
pp. 2835-2847 ◽  
Author(s):  
LEONARDO CASTELLANI
Keyword(s):  
Quantum Groups ◽  
Differential Calculus ◽  
Gauge Theories ◽  
Space Time ◽  
Yang Mills ◽  
Commutation Relations ◽  
Transformation Rules ◽  
Field Strengths

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.

Bicovariant differential calculus on the quantum superspace ℝq(1|2)

2016 ◽  
Vol 15 (09) ◽  
pp. 1650172 ◽  
Author(s):  
Salih Celik
Keyword(s):  
Hopf Algebra ◽  
Differential Calculus ◽  
Vector Fields ◽  
Lie Superalgebra ◽  
Function Algebra ◽  
Algebra Structure ◽  
Hopf Algebra Structure ◽  
Bicovariant Differential Calculus ◽  
Quantum Supergroup ◽  
Dual Hopf Algebra

Super-Hopf algebra structure on the function algebra on the extended quantum superspace has been defined. It is given a bicovariant differential calculus on the superspace. The corresponding (quantum) Lie superalgebra of vector fields and its Hopf algebra structure are obtained. The dual Hopf algebra is explicitly constructed. A new quantum supergroup that is the symmetry group of the differential calculus is found.

Bicovariant differential calculus on quantum groupsSU q (N) andSO q (N)

1991 ◽  
Vol 142 (3) ◽  
pp. 605-641 ◽  
Author(s):  
Ursula Carow-Watamura ◽  
Michael Schlieker ◽  
Satoshi Watamura ◽  
Wolfgang Weich
Keyword(s):  
Differential Calculus ◽  
Bicovariant Differential Calculus

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.

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