Bicovariant differential calculus on the two‐parameter quantum group GLp,q(2)

1992 ◽  
Vol 33 (10) ◽  
pp. 3313-3329 ◽  
Author(s):  
Xiao‐Dong Sun ◽  
Shi‐Kun Wang
Keyword(s):  
Quantum Group ◽  
Differential Calculus ◽  
Bicovariant Differential Calculus ◽  
Two Parameter

A Two-parameter bicovariant differential calculus on the (1 + 2)-dimensional q-superspace

2016 ◽  
Vol 13 (03) ◽  
pp. 1650029
Author(s):  
Ergün Yasar
Keyword(s):  
Hopf Algebra ◽  
Differential Calculus ◽  
Lie Superalgebra ◽  
Point Of View ◽  
Algebra Structure ◽  
Hopf Algebra Structure ◽  
Bicovariant Differential Calculus ◽  
Two Parameter

We construct a two-parameter bicovariant differential calculus on [Formula: see text] with the help of the covariance point of view using the Hopf algebra structure of [Formula: see text]. To achieve this, we first use the consistency of calculus and the approach of [Formula: see text]-matrix which satisfies both ungraded and graded Yang–Baxter equations. In particular, based on this differential calculus, we investigate Cartan–Maurer forms for this [Formula: see text]-superspace. Finally, we obtain the quantum Lie superalgebra corresponding the Cartan–Maurer forms.

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).

scholarly journals CALCULI, HODGE OPERATORS AND LAPLACIANS ON A QUANTUM HOPF FIBRATION

2011 ◽  
Vol 23 (06) ◽  
pp. 575-613 ◽  
Author(s):  
GIOVANNI LANDI ◽  
ALESSANDRO ZAMPINI
Keyword(s):  
Quantum Group ◽  
Homogeneous Space ◽  
Differential Calculus ◽  
Three Dimensional ◽  
Line Bundles ◽  
Hopf Fibration ◽  
Quantum Sphere ◽  
Bicovariant Differential Calculus ◽  
Quantum Homogeneous Space ◽  
Laplacian Operators

We describe Laplacian operators on the quantum group SUq(2) equipped with the four-dimensional bicovariant differential calculus of Woronowicz as well as on the quantum homogeneous space [Formula: see text] with the restricted left covariant three-dimensional differential calculus. This is done by giving a family of Hodge dualities on both the exterior algebras of SUq(2) and [Formula: see text]. We also study gauged Laplacian operators acting on sections of line bundles over the quantum sphere.

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 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.

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