scholarly journals Representations and classification of the compact quantum groups Uq(2) for complex deformation parameters

2021 ◽  
pp. 2150020
Author(s):  
Satyajit Guin ◽  
Bipul Saurabh
Keyword(s):  
Quantum Group ◽  
Jacobi Polynomials ◽  
Compact Quantum Group ◽  
Plancherel Formula ◽  
Irreducible Representations ◽  
Complex Deformation ◽  
Deformation Parameters ◽  
Irreducible Components ◽  
The Matrix ◽  
Compact Quantum Groups

In this paper, we obtain a complete list of inequivalent irreducible representations of the compact quantum group [Formula: see text] for nonzero complex deformation parameters [Formula: see text], which are not roots of unity. The matrix coefficients of these representations are described in terms of the little [Formula: see text]-Jacobi polynomials. The Haar state is shown to be faithful and an orthonormal basis of [Formula: see text] is obtained. Thus, we have an explicit description of the Peter–Weyl decomposition of [Formula: see text]. As an application, we discuss the Fourier transform and establish the Plancherel formula. We also describe the decomposition of the tensor product of two irreducible representations into irreducible components. Finally, we classify the compact quantum group [Formula: see text].

scholarly journals Compact quantum groups and their corepresentations

1998 ◽  
Vol 57 (1) ◽  
pp. 73-91
Author(s):  
Huu Hung Bui
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Duality Theorem ◽  
Compact Quantum Group ◽  
Compact Groups ◽  
Matrix Elements ◽  
Orthogonality Relations ◽  
The Matrix ◽  
Compact Quantum Groups ◽  
Abstract Categories

A compact quantum group is defined to be a unital Hopf C*–algebra generated by the matrix elements of a family of invertible corepresentations. We present a version of the Tannaka–Krein duality theorem for compact quantum groups in the context of abstract categories; this result encompasses the result of Woronowicz and the classical Tannaka-Krein duality theorem. We construct the orthogonality relations (similar to the case of compact groups). The Plancherel theorem is then established.

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 Categorical Aspects of Quantum Groups: Multipliers and Intrinsic Groups

2016 ◽  
Vol 68 (2) ◽  
pp. 309-333 ◽  
Author(s):  
Matthew Daws
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Compact Quantum Group ◽  
Locally Compact ◽  
Multiplier Algebra ◽  
Locally Compact Quantum Groups ◽  
Locally Compact Quantum Group ◽  
Compact Quantum Groups ◽  
Group Form ◽  
Classical Quantum

AbstractWe show that the assignment of the (left) completely bounded multiplier algebra Mlcb(L1()) to a locally compact quantum group , and the assignment of the intrinsic group, form functors between appropriate categories. Morphisms of locally compact quantum groups can be described by Hopf *-homomorphisms between universal C*-algebras, by bicharacters, or by special sorts of coactions. We show that the whole theory of completely bounded multipliers can be lifted to the universal C*-algebra level, and that the different pictures of both multipliers (reduced, universal, and as centralisers) and morphisms then interact in extremely natural ways. The intrinsic group of a quantum group can be realised as a class of multipliers, and so our techniques immediately apply. We also show how to think of the intrinsic group using the universal C*-algebra picture, and then, again, how the differing views on the intrinsic group interact naturally with morphisms. We show that the intrinsic group is the “maximal classical” quantum subgroup of a locally compact quantum group, that it is even closed in the strong Vaes sense, and that the intrinsic group functor is an adjoint to the inclusion functor from locally compact groups to quantum groups.

Amenability and Co-Amenability for Locally Compact Quantum Groups

2003 ◽  
Vol 14 (08) ◽  
pp. 865-884 ◽  
Author(s):  
E. Bédos ◽  
L. Tuset
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Compact Quantum Group ◽  
Locally Compact ◽  
Locally Compact Quantum Groups ◽  
Weak Containment ◽  
Locally Compact Quantum Group ◽  
Compact Quantum Groups

We define concepts of amenability and co-amenability for locally compact quantum groups in the sense of J. Kustermans and S. Vaes. Co-amenability of a lcqg (locally compact quantum group) is proved to be equivalent to a series of statements, all of which imply amenability of the dual lcqg. Further, it is shown that if a lcqg is amenable, then its universal dual lcqg is nuclear. We also define and study amenability and weak containment concepts for representations and corepresentations of lcqg's.

scholarly journals Compact Operators in Regular LCQ Groups

2014 ◽  
Vol 57 (3) ◽  
pp. 546-550 ◽  
Author(s):  
Mehrdad Kalantar
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Compact Operators ◽  
Compact Quantum Group ◽  
Locally Compact ◽  
Locally Compact Quantum Groups ◽  
Locally Compact Quantum Group ◽  
Nikodym Property ◽  
Compact Quantum Groups

AbstractWe show that a regular locally compact quantum group 𝔾 is discrete if and only if 𝓛∞(𝔾) contains non-zero compact operators on 𝓛2(𝔾). As a corollary we classify all discrete quantum groups among regular locally compact quantum groups 𝔾 where 𝓛1(𝔾) has the Radon-Nikodym property.

scholarly journals Generating Functionals for Locally Compact Quantum Groups

2020 ◽  
Author(s):  
Adam Skalski ◽  
Ami Viselter
Keyword(s):  
Quantum Group ◽  
Compact Quantum Group ◽  
Locally Compact ◽  
Generating Functional ◽  
Positive Functional ◽  
C Algebra ◽  
Locally Compact Quantum Groups ◽  
Dense Subalgebra ◽  
Locally Compact Quantum Group ◽  
Compact Quantum Groups

Abstract Every symmetric generating functional of a convolution semigroup of states on a locally compact quantum group is shown to admit a dense unital *-subalgebra with core-like properties in its domain. On the other hand we prove that every normalised, symmetric, hermitian conditionally positive functional on a dense *-subalgebra of the unitisation of the universal C$^*$-algebra of a locally compact quantum group, satisfying certain technical conditions, extends in a canonical way to a generating functional. Some consequences of these results are outlined, notably those related to constructing cocycles out of convolution semigroups.

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 POISSON BOUNDARIES OVER LOCALLY COMPACT QUANTUM GROUPS

2013 ◽  
Vol 24 (03) ◽  
pp. 1350023 ◽  
Author(s):  
MEHRDAD KALANTAR ◽  
MATTHIAS NEUFANG ◽  
ZHONG-JIN RUAN
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Harmonic Functions ◽  
Compact Quantum Group ◽  
Locally Compact ◽  
Locally Compact Quantum Groups ◽  
Compact Quantum Groups ◽  
Concrete Realization ◽  
Separable Case ◽  
Noncommutative Analogue

We present versions of several classical results on harmonic functions and Poisson boundaries in the setting of locally compact quantum groups. In particular, the Choquet–Deny theorem holds for compact quantum groups; also, the result of Kaimanovich–Vershik and Rosenblatt, which characterizes group amenability in terms of harmonic functions, admits a noncommutative analogue in the separable case. We also explore the relation between classical and quantum Poisson boundaries by investigating the spectrum of the quantum group. We apply this machinery to find a concrete realization of the Poisson boundaries of the compact quantum group SU q(2) arising from measures on its spectrum.

scholarly journals Torsion and K-theory for Some Free Wreath Products

2018 ◽  
Vol 2020 (6) ◽  
pp. 1639-1670
Author(s):  
Amaury Freslon ◽  
Rubén Martos
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Free Groups ◽  
Wreath Products ◽  
Compact Quantum Group ◽  
Compact Quantum Groups ◽  
K Theory ◽  
Torsion Free

Abstract We classify torsion actions of free wreath products of arbitrary compact quantum groups by $S_{N}^{+}$ and use this to prove that if $\mathbb{G}$ is a torsion-free compact quantum group satisfying the strong Baum–Connes property then $\mathbb{G}\wr _{\ast }S_{N}^{+}$ also satisfies the strong Baum–Connes property. We then compute the K-theory of free wreath products of classical and quantum free groups by $SO_{q}(3)$.

scholarly journals Locally compact quantum groups in the von Neumann algebraic setting

2003 ◽  
Vol 92 (1) ◽  
pp. 68 ◽  
Author(s):  
Johan Kustermans ◽  
Stefaan Vaes
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Compact Quantum Group ◽  
Locally Compact ◽  
Von Neumann ◽  
Algebraic Setting ◽  
Locally Compact Quantum Groups ◽  
Algebraic Quantum ◽  
Locally Compact Quantum Group ◽  
Compact Quantum Groups

In this paper we complete in several aspects the picture of locally compact quantum groups. First of all we give a definition of a locally compact quantum group in the von Neumann algebraic setting and show how to deduce from it a $C^*$-algebraic quantum group. Further we prove several results about locally compact quantum groups which are important for applications, but were not yet settled in our paper [8]. We prove a serious strengthening of the left invariance of the Haar weight, and we give several formulas connecting the locally compact quantum group with its dual. Loosely speaking we show how the antipode of the locally compact quantum group determines the modular group and modular conjugation of the dual locally compact quantum group.

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