scholarly journals Classification of globally colorized categories of partitions

2018 ◽  
Vol 21 (04) ◽  
pp. 1850029 ◽  
Author(s):  
Daniel Gromada
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Tensor Category ◽  
Set Partitions

Set partitions closed under certain operations form a tensor category. They give rise to certain subgroups of the free orthogonal quantum group [Formula: see text], the so-called easy quantum groups, introduced by Banica and Speicher in 2009. This correspondence was generalized to two-colored set partitions, which, in addition, assign a black or white color to each point of a set. Globally colorized categories of partitions are those categories that are invariant with respect to arbitrary permutations of colors. This paper presents a classification of globally colorized categories. In addition, we show that the corresponding unitary quantum groups can be constructed from the orthogonal ones using tensor complexification.

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 Classification of Integral Modular Categories of Frobenius–Perron Dimension pq4 and p2q2

2014 ◽  
Vol 57 (4) ◽  
pp. 721-734 ◽  
Author(s):  
Paul Bruillard ◽  
Cásar Galindo ◽  
Seung-Moon Hong ◽  
Yevgenia Kashina ◽  
Deepak Naidu ◽  
...  
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Fusion Categories ◽  
Modular Category

AbstractWe classify integral modular categories of dimension pq4 and p2q2, where p and q are distinct primes. We show that such categories are always group-theoretical, except for categories of dimension 4q2. In these cases there are well-known examples of non-group-theoretical categories, coming from centers of Tambara–Yamagami categories and quantum groups. We show that a non-grouptheoretical integral modular category of dimension 4q2 is either equivalent to one of these well-known examples or is of dimension 36 and is twist-equivalent to fusion categories arising froma certain quantum group.

REPRESENTATIONS OF QUANTUM GROUPS AT ROOTS OF 1: REDUCTION TO THE EXCEPTIONAL CASE

1992 ◽  
Vol 07 (supp01a) ◽  
pp. 141-149 ◽  
Author(s):  
CORRADO DE CONCINI ◽  
VICTOR G. KAC
Keyword(s):  
Irreducible Representation ◽  
Quantum Group ◽  
Quantum Groups ◽  
Exceptional Case

This paper is a continuation of the papers [DC-K] and [DC-K-P] on representations of quautum groups at roots of 1. Here we show that an irreducible representation of a quantum group at an odd root of 1 can be uniquely induced from an exceptional representation of a smaller quantum group. This reduces the classification of representations, the calculation of their characters and dimensions, etc, to the exceptional case.

scholarly journals ON SECOND COHOMOLOGY OF DUALS OF COMPACT GROUPS

2011 ◽  
Vol 22 (09) ◽  
pp. 1231-1260 ◽  
Author(s):  
SERGEY NESHVEYEV ◽  
LARS TUSET
Keyword(s):  
Quantum Groups ◽  
Hopf Algebras ◽  
Cohomology Group ◽  
Tensor Category ◽  
Compact Groups ◽  
Tensor Categories ◽  
Connected Group ◽  
Compact Quantum Groups ◽  
Algebraic Counterpart

We show that for any compact connected group G the second cohomology group defined by unitary invariant two-cocycles on Ĝ is canonically isomorphic to [Formula: see text]. This implies that the group of autoequivalences of the C*-tensor category Rep G is isomorphic to [Formula: see text]. We also show that a compact connected group G is completely determined by Rep G. More generally, extending a result of Etingof–Gelaki and Izumi–Kosaki we describe all pairs of compact separable monoidally equivalent groups. The proofs rely on the theory of ergodic actions of compact groups developed by Landstad and Wassermann and on its algebraic counterpart developed by Etingof and Gelaki for the classification of triangular semisimple Hopf algebras. We give a self-contained account of amenability of tensor categories, fusion rings and discrete quantum groups, and prove an analog of Radford's theorem on minimal Hopf subalgebras of quasitriangular Hopf algebras for compact quantum groups.

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 On the classification of easy quantum groups

2013 ◽  
Vol 245 ◽  
pp. 500-533 ◽  
Author(s):  
Moritz Weber
Keyword(s):  
Quantum Groups

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.

HYBRID-TYPE MODEL AND THE DIRECT SUM OF QUANTUM GROUPS

1991 ◽  
Vol 06 (13) ◽  
pp. 1177-1183 ◽  
Author(s):  
TETSUO DEGUCHI ◽  
AKIRA FUJII
Keyword(s):  
Quantum Group ◽  
Inverse Scattering ◽  
Quantum Groups ◽  
Group Structure ◽  
Formal Group ◽  
Inverse Scattering Method ◽  
Type Model ◽  
Scattering Method ◽  
Direct Sum ◽  
Hybrid Type

We present the quantum formal group derived from the hybrid-type model. The quantum group structure is given by the direct sum of several quantum groups. We show that by applying the quantum inverse scattering method to the direct sum of the several quantum groups we can reconstruct the hybrid-type model.

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