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 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).

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 The combinatorics of an algebraic class of easy quantum groups

2014 ◽  
Vol 17 (03) ◽  
pp. 1450016 ◽  
Author(s):  
Sven Raum ◽  
Moritz Weber
Keyword(s):  
Symmetric Group ◽  
Quantum Group ◽  
Permutation Group ◽  
Free Product ◽  
Quantum Groups ◽  
Orthogonal Group ◽  
Point Of View ◽  
Algebraic Point ◽  
Matrix Quantum

Easy quantum groups are compact matrix quantum groups, whose intertwiner spaces are given by the combinatorics of categories of partitions. This class contains the symmetric group Sn and the orthogonal group On as well as Wang's quantum permutation group [Formula: see text] and his free orthogonal quantum group [Formula: see text]. In this paper, we study a particular class of categories of partitions to each of which we assign a subgroup of the infinite free product of the cyclic group of order two. This is an important step in the classification of all easy quantum groups and we deduce that there are uncountably many of them. We focus on the combinatorial aspects of this assignment, complementing the quantum algebraic point of view presented in another paper.

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.

Duality for the matrix quantum group GLp,q(2,C)

1992 ◽  
Vol 33 (10) ◽  
pp. 3419-3430 ◽  
Author(s):  
V. K. Dobrev
Keyword(s):  
Quantum Group ◽  
The Matrix ◽  
Matrix Quantum

scholarly journals UNIVERSAL QUANTUM GROUPS

1996 ◽  
Vol 07 (02) ◽  
pp. 255-263 ◽  
Author(s):  
ALFONS VAN DAELE ◽  
SHUZHOU WANG
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Universal Quantum ◽  
Group A ◽  
Matrix Quantum

For each invertible m×m matrix Q a compact matrix quantum group Au(Q) is constructed. These quantum groups are shown to be universal in the sense that any compact matrix quantum group is a quantum subgroup of some of them. Their orthogonal version Ao(Q) is also constructed. Finally, we discuss related constructions in the literature.

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 Introduction to Quantum Groups

1998 ◽  
Vol 10 (04) ◽  
pp. 511-551 ◽  
Author(s):  
P. Podleś ◽  
E. Müller
Keyword(s):  
Hopf Algebra ◽  
Quantum Groups ◽  
Theory Theory ◽  
Topological Quantum ◽  
Algebra Theory ◽  
Basic Facts ◽  
Quantum Spheres ◽  
Real Forms ◽  
Matrix Quantum

We give an elementary introduction to the theory of algebraic and topological quantum groups (in the spirit of S. L. Woronowicz). In particular, we recall the basic facts from Hopf (*-) algebra theory, theory of compact (matrix) quantum groups and the theory of their actions on compact quantum spaces. We also provide the most important examples, including the classification of quantum SL(2)-groups, their real forms and quantum spheres. We also consider quantum SLq(N)-groups and quantum Lorentz groups.

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.

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