DIFFERENTIAL CALCULUS ON INHOMOGENEOUS QUANTUM GROUPS

2000 ◽  
Vol 14 (22n23) ◽  
pp. 2299-2303
Author(s):  
DANIEL DIDT
Keyword(s):  
Hopf Algebra ◽  
Quantum Groups ◽  
Differential Calculus ◽  
Braided Hopf Algebra ◽  
Inhomogeneous Quantum

We investigate the question of covariant differential calculi on the bosonisation of a coquasitriangular Hopf algebra and an associated braided Hopf algebra. As a result we present a general way of obtaining such calculi on inhomogeneous quantum groups.

Double-bosonization of braided groups and the construction of Uq(g)

1999 ◽  
Vol 125 (1) ◽  
pp. 151-192 ◽  
Author(s):  
S. MAJID
Keyword(s):  
Hopf Algebra ◽  
Quantum Groups ◽  
Cartan Subalgebra ◽  
Positive Root ◽  
Point Of View ◽  
Root Space ◽  
Quantum Double ◽  
Braided Group ◽  
Negative Root ◽  
Quasitriangular Hopf Algebra

We introduce a quasitriangular Hopf algebra or ‘quantum group’ U(B), the double-bosonization, associated to every braided group B in the category of H-modules over a quasitriangular Hopf algebra H, such that B appears as the ‘positive root space’, H as the ‘Cartan subalgebra’ and the dual braided group B* as the ‘negative root space’ of U(B). The choice B=Uq(n+) recovers Lusztig's construction of Uq(g); other choices give more novel quantum groups. As an application, our construction provides a canonical way of building up quantum groups from smaller ones by repeatedly extending their positive and negative root spaces by linear braided groups; we explicitly construct Uq(sl3) from Uq(sl2) by this method, extending it by the quantum-braided plane. We provide a fundamental representation of U(B) in B. A projection from the quantum double, a theory of double biproducts and a Tannaka–Krein reconstruction point of view are also provided.

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.

scholarly journals A categorical reconstruction of crystals and quantum groups at q = 0

2019 ◽  
Vol 70 (3) ◽  
pp. 895-925
Author(s):  
Craig Smith
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Lie Algebra ◽  
Quantum Groups ◽  
Analogous Result ◽  
Crystal Bases ◽  
Direct Sums ◽  
Finite Dimensional ◽  
Monoidal Structure

Abstract The quantum co-ordinate algebra Aq(g) associated to a Kac–Moody Lie algebra g forms a Hopf algebra whose comodules are direct sums of finite-dimensional irreducible Uq(g) modules. In this paper, we investigate whether an analogous result is true when q=0. We classify crystal bases as coalgebras over a comonadic functor on the category of pointed sets and encode the monoidal structure of crystals into a bicomonadic structure. In doing this, we prove that there is no coalgebra in the category of pointed sets whose comodules are equivalent to crystal bases. We then construct a bialgebra over Z whose based comodules are equivalent to crystals, which we conjecture is linked to Lusztig’s quantum group at v=∞.

scholarly journals On the Structure of Inhomogeneous Quantum Groups

1997 ◽  
Vol 185 (2) ◽  
pp. 325-358 ◽  
Author(s):  
P. Podleś ◽  
S. L. Woronowicz
Keyword(s):  
Quantum Groups ◽  
Inhomogeneous Quantum

scholarly journals Quantum groups and conformal field theories

1989 ◽  
Vol 329 (1605) ◽  
pp. 343-347
Keyword(s):  
Hopf Algebra ◽  
Quantum Groups ◽  
Hopf Algebras ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Conformal Theory ◽  
Conformal Field

Rational conformal field theories can be interpreted as defining quasi-triangular Hopf algebras. The Hopf algebra is determined by the duality properties of the conformal theory.

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.

scholarly journals A FINITE QUANTUM SYMMETRY OF $M(3, {\Bbb C})$

1998 ◽  
Vol 13 (24) ◽  
pp. 4147-4161 ◽  
Author(s):  
LUDWIK DABROWSKI ◽  
FABRIZIO NESTI ◽  
PASQUALE SINISCALCO
Keyword(s):  
Standard Model ◽  
Exact Sequence ◽  
Hopf Algebra ◽  
Quantum Group ◽  
Recent Work ◽  
Quantum Groups ◽  
Matrix Algebra ◽  
Group Symmetry ◽  
The Standard Model ◽  
The Matrix

The 27-dimensional Hopf algebra A(F), defined by the exact sequence of quantum groups [Formula: see text], [Formula: see text], is studied as a finite quantum group symmetry of the matrix algebra [Formula: see text], describing the color sector of Alain Connes' formulation of the Standard Model. The duality with the Hopf algebra ℋ, investigated in a recent work by Robert Coquereaux, is established and used to define a representation of ℋ on [Formula: see text] and two commuting representation of ℋ on A(F).

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