scholarly journals The quantum disk is not a quantum group

2021 ◽  
pp. 1-11
Author(s):  
Jacek Krajczok ◽  
Piotr M. Sołtan
Keyword(s):  
Quantum Group ◽  
Hilbert Spaces ◽  
Group Structure ◽  
Compact Quantum Group ◽  
Quantum Space ◽  
Quantum Disk

In this paper, we show that the quantum disk, i.e. the quantum space corresponding to the Toeplitz [Formula: see text]-algebra does not admit any compact quantum group structure. We prove that if such a structure existed the resulting compact quantum group would simultaneously be of Kac type and not of Kac type. The main tools used in the solution come from the theory of operators on Hilbert spaces.

scholarly journals ON QUANTUM SEMIGROUP ACTIONS ON FINITE QUANTUM SPACES

2009 ◽  
Vol 12 (03) ◽  
pp. 503-509 ◽  
Author(s):  
PIOTR MIKOŁAJ SOŁTAN
Keyword(s):  
Quantum Group ◽  
Compact Quantum Group ◽  
Quantum Space ◽  
Bohr Compactification ◽  
Continuous Action ◽  
Semigroup Actions ◽  
Finite Dimensional ◽  
Faithful State ◽  
Quantum Group Actions

We show that a continuous action of a quantum semigroup [Formula: see text] on a finite quantum space (finite dimensional C*-algebra) preserving a faithful state comes from a continuous action of the quantum Bohr compactification [Formula: see text] of [Formula: see text]. Using the classification of continuous compact quantum group actions on M2, we give a complete description of all continuous quantum semigroup actions on this quantum space preserving a faithful state.

scholarly journals REGULAR OBJECTS, MULTIPLICATIVE UNITARIES AND CONJUGATION

2002 ◽  
Vol 13 (06) ◽  
pp. 625-665 ◽  
Author(s):  
C. PINZARI ◽  
J. E. ROBERTS
Keyword(s):  
Quantum Group ◽  
Hilbert Spaces ◽  
Compact Quantum Group ◽  
Locally Compact ◽  
C Algebras ◽  
Regular Object ◽  
Left Regular ◽  
Locally Compact Quantum Group ◽  
Left And Right ◽  
Tensor Functor

The notion of left (respectively right) regular object of a tensor C*-category equipped with a faithful tensor functor into the category of Hilbert spaces is introduced. If such a category has a left (respectively right) regular object, it can be interpreted as a category of corepresentations (respectively representations) of some multiplicative unitary. A regular object is an object of the category which is at the same time left and right regular in a coherent way. A category with a regular object is endowed with an associated standard braided symmetry. Conjugation is discussed in the context of multiplicative unitaries and their associated Hopf C*-algebras. It is shown that the conjugate of a left regular object is a right regular object in the same category. Furthermore the representation category of a locally compact quantum group has a conjugation. The associated multiplicative unitary is a regular object in that category.

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]).

INTEGRABLE MODELS OF FIELD THEORY AND SCATTERING ON QUANTUM HYPERBOLOIDS

1992 ◽  
Vol 07 (05) ◽  
pp. 441-446 ◽  
Author(s):  
A. ZABRODIN
Keyword(s):  
Field Theory ◽  
Symmetric Space ◽  
Quantum Group ◽  
Wave Scattering ◽  
Scattering Problem ◽  
Free Particle ◽  
Compact Quantum Group ◽  
Integrable Models ◽  
S Wave

We consider the scattering of two dressed excitations in the antiferromagnetic XXZ spin-1/2 chain and show that it is equivalent to the S-wave scattering problem for a free particle on the certain quantum symmetric space “quantum hyperboloid” related to the non-compact quantum group SU q (1, 1).

POISSON–LIE GROUPS AND CLASSICAL W-ALGEBRAS

1992 ◽  
Vol 07 (05) ◽  
pp. 853-876 ◽  
Author(s):  
V. A. FATEEV ◽  
S. L. LUKYANOV
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Quantum Group ◽  
Lie Groups ◽  
Group Structure ◽  
Two Dimensional ◽  
Poisson Structures ◽  
Additional Symmetries ◽  
Conformal Field ◽  
Poisson Lie Groups

This is the first part of a paper studying the quantum group structure of two-dimensional conformal field theory with additional symmetries. We discuss the properties of the Poisson structures possessing classical W-invariance. The Darboux variables for these Poisson structures are constructed.

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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