ON A HOPF ALGEBRA RELATED TO THE GLq(2) AND GLp,q(2) QUANTUM GROUPS

1996 ◽  
Vol 11 (04) ◽  
pp. 715-732
Author(s):  
B. BASU-MALLICK
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Quantum Groups ◽  
Nonlinear Realization ◽  
Intimate Connection ◽  
Deformation Parameters ◽  
Two Parameter

By adding one extra generator with the standard GL q(2) quantum group, we construct a Hopf algebra [Formula: see text] which depends on two deformation parameters and five generators. Curiously, it turns out that there exists a nonlinear realization of the two-parameter deformed GL p,q(2) quantum group through generators of this [Formula: see text] algebra. Subsequently, we find the invariant noncommutative planes associated with the [Formula: see text] quantum group and also discuss how the well-known Manin planes corresponding to the GL p,q(2) quantum group can be produced automatically, through such construction. Finally, we consider the “colored” extension of the GL p,q(2) quantum group as well as corresponding noncommutative planes and explore their intimate connection with the “colored” extension of [Formula: see text] Hopf structure.

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 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).

ON MATRIX QUANTUM GROUPS OF TYPE An

2000 ◽  
Vol 11 (09) ◽  
pp. 1115-1146 ◽  
Author(s):  
HO Hai PHUNG
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Quantum Groups ◽  
Semi Group ◽  
The Matrix ◽  
Hecke Symmetry ◽  
Matrix Quantum

Given a Hecke symmetry R, one can define a matrix bialgebra ER and a matrix Hopf algebra HR, which are called function rings on the matrix quantum semi-group and matrix quantum groups associated to R. We show that for an even Hecke symmetry, the rational representations of the corresponding quantum group are absolutely reducible and that the fusion coefficients of simple representations depend only on the rank of the Hecke symmetry. Further we compute the quantum rank of simple representations. We also show that the quantum semi-group is "Zariski" dense in the quantum group. Finally we give a formula for the integral.

scholarly journals Quantum E(2) groups for complex deformation parameters

2021 ◽  
pp. 2150021
Author(s):  
Atibur Rahaman ◽  
Sutanu Roy
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Locally Compact ◽  
Complex Deformation ◽  
Circle Group ◽  
Deformation Parameters ◽  
Quantum Analogue ◽  
Group Contraction ◽  
Contraction Procedure ◽  
Matrix Formula

We construct a family of [Formula: see text] deformations of E(2) group for nonzero complex parameters [Formula: see text] as locally compact braided quantum groups over the circle group [Formula: see text] viewed as a quasitriangular quantum group with respect to the unitary [Formula: see text]-matrix [Formula: see text] for all [Formula: see text]. For real [Formula: see text], the deformation coincides with Woronowicz’s [Formula: see text] groups. As an application, we study the braided analogue of the contraction procedure between [Formula: see text] and [Formula: see text] groups in the spirit of Woronowicz’s quantum analogue of the classic Inönü–Wigner group contraction. Consequently, we obtain the bosonization of braided [Formula: see text] groups by contracting [Formula: see text] groups.

scholarly journals BICROSSPRODUCTS OF ALGEBRAIC QUANTUM GROUPS

2013 ◽  
Vol 24 (01) ◽  
pp. 1250131
Author(s):  
L. DELVAUX ◽  
A. VAN DAELE ◽  
S. H. WANG
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Quantum Groups ◽  
Smash Product ◽  
Module Algebra ◽  
Algebraic Quantum ◽  
Algebra Theory ◽  
Extra Structure

Let A and B be two algebraic quantum groups. Assume that B is a right A-module algebra and that A is a left B-comodule coalgebra. If the action and coaction are matched, it is possible to define a coproduct Δ# on the smash product A#B making the pair (A#B, Δ#) into an algebraic quantum group. In this paper we study the various data of the bicrossproduct A#B, such as the modular automorphisms, the modular elements, … and we obtain formulas in terms of the data of the components A and B. Secondly, we look at the dual of A#B (in the sense of algebraic quantum groups) and we show it is itself a bicrossproduct (of the second type) of the duals [Formula: see text] and [Formula: see text]. We give some examples that are typical for algebraic quantum groups. In particular, we focus on the extra structure, provided by the integrals and associated objects. It should be mentioned that with examples of bicrossproducts of algebraic quantum groups, we do get examples that are essentially different from those commonly known in Hopf algebra theory.

The dual of a certain left quantum group

2016 ◽  
Vol 15 (04) ◽  
pp. 1650072 ◽  
Author(s):  
Uma N. Iyer ◽  
Earl J. Taft
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Quantum Groups ◽  
Hopf Algebras ◽  
Current Work ◽  
Independent Interest ◽  
Natural Choice

The second author and his collaborators, Lauve and Rodriguez-Romo (see [A class of left quantum groups modeled after SL[Formula: see text]([Formula: see text]), J. Pure Appl. Algebra 208(3) (2007) 797–803; A left quantum group, J. Algebra 286 (2005) 154–160), have constructed left Hopf algebras which are not Hopf algebras modeled after [Formula: see text]. In particular, they constructed the left quantum group [Formula: see text], along with an epimorphism to the quantum group [Formula: see text], the latter being a Hopf algebra. The current work began as a search for a left Hopf algebra (which is not a Hopf algebra) containing the quantum group [Formula: see text], the latter being a Hopf algebra. The natural choice was to look for the dual of [Formula: see text]. We show that the Hopf dual of [Formula: see text] is equal to the Hopf dual of [Formula: see text], which is of independent interest. Thus the Hopf dual of [Formula: see text] is a Hopf algebra. The original search of a left Hopf algebra which is not a Hopf algebra, larger than [Formula: see text], is still open.

scholarly journals $L^2$-homology for compact quantum groups

2008 ◽  
Vol 103 (1) ◽  
pp. 111 ◽  
Author(s):  
David Kyed
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Quantum Groups ◽  
Lie Group ◽  
Betti Numbers ◽  
Compact Quantum Group ◽  
Discrete Groups ◽  
Compact Lie Group ◽  
Matrix Coefficients ◽  
Compact Quantum Groups

A notion of $L^2$-homology for compact quantum groups is introduced, generalizing the classical notion for countable, discrete groups. If the compact quantum group in question has tracial Haar state, it is possible to define its $L^2$-Betti numbers and Novikov-Shubin invariants/capacities. It is proved that these $L^2$-Betti numbers vanish for the Gelfand dual of a compact Lie group and that the zeroth Novikov-Shubin invariant equals the dimension of the underlying Lie group. Finally, we relate our approach to the approach of A. Connes and D. Shlyakhtenko by proving that the $L^2$-Betti numbers of a compact quantum group, with tracial Haar state, are equal to the Connes-Shlyakhtenko $L^2$-Betti numbers of its Hopf $*$-algebra of matrix coefficients.

Compact Hopf *-Algebras, Quantum Enveloping Algebras and Dual Woronowicz Algebras for Quantum Lorentz Groups

1997 ◽  
Vol 08 (07) ◽  
pp. 959-997 ◽  
Author(s):  
Hideki Kurose ◽  
Yoshiomi Nakagami
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Quantum Groups ◽  
Hopf Algebras ◽  
Canonical Representation ◽  
Compact Quantum Group ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Quantum Enveloping Algebra ◽  
Definition Of

A compact Hopf *-algebra is a compact quantum group in the sense of Koornwinder. There exists an injective functor from the category of compact Hopf *-algebras to the category of compact Woronowicz algebras. A definition of the quantum enveloping algebra Uq(sl(n,C)) is given. For quantum groups SUq(n) and SLq(n,C), the commutant of a canonical representation of the quantum enveloping algebra for q coincides with the commutant of the dual Woronowicz algebra for q-1.

scholarly journals RIESZ TRANSFORMS ON COMPACT QUANTUM GROUPS AND STRONG SOLIDITY

2021 ◽  
pp. 1-37
Author(s):  
Martijn Caspers
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Riesz Transform ◽  
Wreath Products ◽  
Compact Quantum Group ◽  
Riesz Transforms ◽  
Markov Semigroup ◽  
Markov Semigroups ◽  
Free Products ◽  
Compact Quantum Groups

Abstract One of the main aims of this paper is to give a large class of strongly solid compact quantum groups. We do this by using quantum Markov semigroups and noncommutative Riesz transforms. We introduce a property for quantum Markov semigroups of central multipliers on a compact quantum group which we shall call ‘approximate linearity with almost commuting intertwiners’. We show that this property is stable under free products, monoidal equivalence, free wreath products and dual quantum subgroups. Examples include in particular all the (higher-dimensional) free orthogonal easy quantum groups. We then show that a compact quantum group with a quantum Markov semigroup that is approximately linear with almost commuting intertwiners satisfies the immediately gradient- ${\mathcal {S}}_2$ condition from [10] and derive strong solidity results (following [10]). Using the noncommutative Riesz transform we also show that these quantum groups have the Akemann–Ostrand property; in particular, the same strong solidity results follow again (now following [27]).

Invariants of 3-manifolds derived from universal invariants of framed links

1995 ◽  
Vol 117 (2) ◽  
pp. 259-273 ◽  
Author(s):  
Tomotada Ohtsuki
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Independent Method ◽  
Topological Invariant ◽  
Finite Dimensional

Reshetikhin and Turaev [10] gave a method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra (e.g. a quantum group Uq(sl2)) using finite-dimensional representations of it. In this paper we give another independent method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra via universal invariants of framed links without using representations of the algebra. For Uq(sl2) these two methods give different invariants of 3-manifolds.

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