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 Real Forms of the Fractional Supergroup SL(2,C)

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 933
Author(s):  
Yasemen Ucan ◽  
Resat Kosker
Keyword(s):  
Quantum Group ◽  
Lie Group ◽  
The Real ◽  
And Mathematics ◽  
Real Forms

The real forms of complex groups (or algebras) are important in physics and mathematics. The Lie group SL2,C is one of these important groups. There are real forms of the classical Lie group SL2,C and the quantum group SL2,C in the literature. Inspired by this, in our study, we obtain the real forms of the fractional supergroups shown with A3NSL2,C, for the non-trivial N = 1 and N = 2 cases, that is, the real forms of the fractional supergroups A31SL2,C and A32SL2,C.

Quantum Homomorphisms of Associative Algebras

1997 ◽  
Vol 12 (38) ◽  
pp. 2963-2974
Author(s):  
A. E. F. Djemai
Keyword(s):  
Quantum Group ◽  
Associative Algebra ◽  
Associative Algebras ◽  
Type Formula ◽  
Algebra Of Functions ◽  
Inhomogeneous Quantum

Given an associative algebra A generated by {ek, k=1, 2,…} and with an internal law of type: [Formula: see text], we first show that it is possible to construct a quantum bi-algebra [Formula: see text] with unit and generated by (non-necessarily commutative) elements [Formula: see text] satisfying the relations: [Formula: see text]. This leads one to define a quantum homomorphism[Formula: see text]. We then treat the example of the algebra of functions on a set of N elements and we show, for the case N=2, that the resulting bihyphen;algebra is an inhomogeneous quantum group. We think that this method can be used to construct quantum inhomogeneous groups.

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.

Sign in / Sign up

Export Citation Format

Share Document