scholarly journals Probabilistic boundaries of finite extensions of quantum groups

2017 ◽  
Vol 20 (04) ◽  
pp. 1750026 ◽  
Author(s):  
Sara Malacarne ◽  
Sergey Neshveyev
Keyword(s):  
Random Walk ◽  
Harmonic Function ◽  
Quantum Group ◽  
Quantum Groups ◽  
Tensor Categories ◽  
Martin Boundaries ◽  
Exact Sequences

Given a discrete quantum group [Formula: see text] with a finite normal quantum subgroup [Formula: see text], we show that any positive, possibly unbounded, harmonic function on [Formula: see text] with respect to an irreducible invariant random walk is [Formula: see text]-invariant. This implies that, under suitable assumptions, the Poisson and Martin boundaries of [Formula: see text] coincide with those of [Formula: see text]. A similar result is also proved in the setting of exact sequences of C[Formula: see text]-tensor categories. As an immediate application, we conclude that the boundaries of the duals of the group-theoretical easy quantum groups are classical.

scholarly journals Martin boundaries of the duals of free unitary quantum groups

2019 ◽  
Vol 155 (6) ◽  
pp. 1171-1193
Author(s):  
Sara Malacarne ◽  
Sergey Neshveyev
Keyword(s):  
Random Walk ◽  
Quantum Group ◽  
Quantum Groups ◽  
Free Group ◽  
Cayley Tree ◽  
Martin Boundary ◽  
Finite Range ◽  
Quantum Random Walk ◽  
Martin Boundaries ◽  
Topological Boundary

Given a free unitary quantum group $G=A_{u}(F)$, with $F$ not a unitary $2\times 2$ matrix, we show that the Martin boundary of the dual of $G$ with respect to any $G$-${\hat{G}}$-invariant, irreducible, finite-range quantum random walk coincides with the topological boundary defined by Vaes and Vander Vennet. This can be thought of as a quantum analogue of the fact that the Martin boundary of a free group coincides with the space of ends of its Cayley tree.

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.

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.

scholarly journals DOUBLE CROSSED PRODUCTS OF LOCALLY COMPACT QUANTUM GROUPS

2005 ◽  
Vol 4 (1) ◽  
pp. 135-173 ◽  
Author(s):  
Saad Baaj ◽  
Stefaan Vaes
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Matched Pair ◽  
Crossed Products ◽  
Locally Compact ◽  
Locally Compact Quantum Groups ◽  
Algebraic Properties ◽  
Locally Compact Quantum Group ◽  
Compact Quantum Groups

For a matched pair of locally compact quantum groups, we construct the double crossed product as a locally compact quantum group. This construction generalizes Drinfeld’s quantum double construction. We study the modular theory and the $\mathrm{C}^*$-algebraic properties of these double crossed products, as well as several links between double crossed products and bicrossed products. In an appendix, we study the Radon–Nikodym derivative of a weight under a quantum group action (following Yamanouchi) and obtain, as a corollary, a new characterization of closed quantum subgroups. AMS 2000 Mathematics subject classification: Primary 46L89. Secondary 46L65

scholarly journals Subgroups of quantum groups

2018 ◽  
Vol 70 (2) ◽  
pp. 509-533
Author(s):  
Georgia Christodoulou
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Homogeneous Spaces ◽  
General Definition ◽  
Monoidal Categories ◽  
Categorical Approach ◽  
Quantum Homogeneous Spaces

Abstract We investigate the notion of a subgroup of a quantum group. We suggest a general definition, which takes into account the work that has been done for quantum homogeneous spaces. We further restrict our attention to reductive subgroups, where some faithful flatness conditions apply. Furthermore, we proceed with a categorical approach to the problem of finding quantum subgroups. We translate all existing results into the language of module and monoidal categories and give another characterization of the notion of a quantum subgroup.

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 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.

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