Classification of bicovariant differential calculi
Shahn Majid,Shahn Majid +1 more
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In this paper, the bicovariant first-order differential calculi on a factorisable quantum group with the Peter-Weyl decomposition property are in 1−1 correspondence with irreducible representations V of the quantum group enveloping algebra.About:
This article is published in Journal of Geometry and Physics.The article was published on 1998-04-01 and is currently open access. It has received 65 citations till now. The article focuses on the topics: Quantum group & Quantum differential calculus.read more
Citations
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Quantum groups and noncommutative geometry
TL;DR: Quantum groups emerged in the latter quarter of the 20th century as, on the one hand, a deep and natural generalization of symmetry groups for certain integrable systems, and on the other as part of a generalisation of geometry itself powerful enough to make sense in the quantum domain this paper.
Journal ArticleDOI
Gravity induced from quantum spacetime
Edwin J. Beggs,Shahn Majid +1 more
TL;DR: In this article, it was shown that tensoriality constraints in non-commutative Riemannian geometry in the two-dimensional bicrossproduct model quantum spacetime algebra [x, t] = λx drastically reduce the moduli of possible metrics g up to normalization to a single real parameter.
Journal ArticleDOI
Riemannian geometry of quantum groups and finite groups with nonuniversal differentials
TL;DR: In this article, a non-commutative Riemannian manifold was constructed on dual quasitriangular Hopf algebras with the standard bicovariant differential calculus.
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Semiclassical differential structures
Edwin J. Beggs,Shahn Majid +1 more
TL;DR: In this paper, the standard notion of differential calculus in noncommutative geometry on algebras and quantum groups is considered, and the moduli space of bicovariant infinitesimal data for quasitriangular Poisson?Lie groups has a canonical reference point which is flat in the triangular case.
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Meaning of Noncommutative Geometry and the Planck-Scale Quantum Group
TL;DR: In this article, a non-commutative geometric approach to Planck scale physics coming out of quantum groups is introduced, and the general meaning of noncommutativity of position space as potentially a new force in Nature is explained as equivalent under quantum group Fourier transform to curvature in momentum space.
References
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Book
Quantum Groups
TL;DR: In this paper, the authors introduce the theory of quantum groups with emphasis on the spectacular connections with knot theory and Drinfeld's recent fundamental contributions and present the quantum groups attached to SL2 as well as the basic concepts of the Hopf algebras.
Book
Foundations of Quantum Group Theory
TL;DR: In this paper, the authors define Hopf algebras as "quasitriangular Hopf-algebraes" and introduce matrix quantum groups and bicrossproduct hopf alges.
Journal ArticleDOI
Differential calculus on compact matrix pseudogroups (quantum groups)
TL;DR: In this paper, a general theory of non-commutative differential geometry on quantum groups is developed, where bicovariant bimodules as objects analogous to tensor bundles over Lie groups are studied.
Journal ArticleDOI
Quantum group gauge theory on quantum spaces
Tomasz Brzeziński,Shahn Majid +1 more
TL;DR: In this paper, the authors constructed quantum group-valued canonical connections on quantum homogeneous spaces, including aq-deformed Dirac monopole on the quantum sphere of Podles with quantum differential structure coming from the 3D calculus of Woronowicz on quantum spaces.
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Quantum R-matrices and factorization problems
TL;DR: In this paper, a relation between quantum R-matrices and certain factorization problems in Hopf algebras is established, and a definition of dressinf transformation in the quantum case is also given.