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.

scholarly journals LOCALLY COMPACT QUANTUM GROUPS IN THE UNIVERSAL SETTING

2001 ◽  
Vol 12 (03) ◽  
pp. 289-338 ◽  
Author(s):  
JOHAN KUSTERMANS
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Group Structure ◽  
Locally Compact ◽  
Bijective Correspondence ◽  
Locally Compact Quantum Groups ◽  
Algebraic Quantum ◽  
Compact Quantum Groups ◽  
Group A ◽  
Fine Tune

In this paper we associate to every reduced C *-algebraic quantum group (A, Δ) (as defined in [11]) a universal C *-algebraic quantum group (Au, Δu). We fine tune a proof of Kirchberg to show that every *-representation of a modified L 1-space is generated by a unitary corepresentation. By taking the universal enveloping C *-algebra of a dense sub *-algebra of A we arrive at the C *-algebra Au. We show that this C *-algebra Au carries a quantum group structure which is a rich as its reduced companion. We also establish a bijective correspondence between quantum group morphisms and certain co-actions.

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 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 MONOIDS, EMBEDDING FUNCTORS AND QUANTUM GROUPS

2008 ◽  
Vol 19 (01) ◽  
pp. 93-123 ◽  
Author(s):  
MICHAEL MÜGER ◽  
LARS TUSET
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Regular Representation ◽  
Tensor Category ◽  
Left Regular Representation ◽  
Left Regular ◽  
Group A ◽  
Absorbing Property

We show that the left regular representation πl of a discrete quantum group (A, Δ) has the absorbing property and forms a monoid [Formula: see text] in the representation category Rep (A, Δ). Next we show that an absorbing monoid in an abstract tensor *-category [Formula: see text] gives rise to an embedding functor (or fiber functor) [Formula: see text], and we identify conditions on the monoid, satisfied by [Formula: see text], implying that E is *-preserving. As is well-known, from an embedding functor [Formula: see text] the generalized Tannaka theorem produces a discrete quantum group (A, Δ) such that [Formula: see text]. Thus, for a C*-tensor category [Formula: see text] with conjugates and irreducible unit the following are equivalent: (1) [Formula: see text] is equivalent to the representation category of a discrete quantum group (A, Δ), (2) [Formula: see text] admits an absorbing monoid, (3) there exists a *-preserving embedding functor [Formula: see text].

scholarly journals C*-Algebraic Quantum Groups Arising from Algebraic Quantum Groups

1997 ◽  
Vol 08 (08) ◽  
pp. 1067-1139 ◽  
Author(s):  
J. Kustermans ◽  
A. van Daele
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Algebraic Quantum ◽  
Group A ◽  
Definition Of

We associate to an algebraic quantum group a C*-algebraic quantum group and show that this C*-algebraic quantum group essentially satisfies an upcoming definition of Masuda, Nakagami and Woronowicz.

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]).

scholarly journals From Quantum Groups to Groups

2013 ◽  
Vol 65 (5) ◽  
pp. 1073-1094 ◽  
Author(s):  
Mehrdad Kalantar ◽  
Matthias Neufang
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Locally Compact ◽  
Large Classes ◽  
Locally Compact Quantum Groups ◽  
Locally Compact Quantum Group ◽  
Recent Developments ◽  
Compact Quantum Groups ◽  
Quantum Version ◽  
Quantum Point

AbstractIn this paper we use the recent developments in the representation theory of locally compact quantum groups, to assign to each locally compact quantum group 𝔾 a locally compact group 𝔾˜ that is the quantum version of point-masses and is an invariant for the latter. We show that “quantum point-masses” can be identified with several other locally compact groups that can be naturally assigned to the quantum group 𝔾. This assignment preserves compactness as well as discreteness (hence also finiteness), and for large classes of quantum groups, amenability. We calculate this invariant for some of the most well-known examples of non-classical quantum groups. Also, we show that several structural properties of 𝔾 are encoded by 𝔾˜; the latter, despite being a simpler object, can carry very important information about 𝔾.

scholarly journals Strong Asymptotic Freeness for Free Orthogonal Quantum Groups

2014 ◽  
Vol 57 (4) ◽  
pp. 708-720 ◽  
Author(s):  
Michael Brannan
Keyword(s):  
Strong Convergence ◽  
Quantum Group ◽  
Quantum Groups ◽  
Convergence Theorem ◽  
Convergence Result ◽  
Operator Norm ◽  
Matrix Coefficient ◽  
Strong Convergence Theorem ◽  
Convergence Results ◽  
Noncommutative Polynomials

AbstractIt is known that the normalized standard generators of the free orthogonal quantum groupO+Nconverge in distribution to a free semicircular system as N → ∞. In this note, we substantially improve this convergence result by proving that, in addition to distributional convergence, the operator normof any non-commutative polynomial in the normalized standard generators ofO+Nconverges asN→ ∞ to the operator norm of the corresponding non-commutative polynomial in a standard free semicircular system. Analogous strong convergence results are obtained for the generators of free unitary quantum groups. As applications of these results, we obtain a matrix-coefficient version of our strong convergence theorem, and we recover a well-knownL2-L∞norm equivalence for noncommutative polynomials in free semicircular systems.

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