Understanding the q-Factors in Quantum Group Symmetry

1997 ◽  
Vol 52 (1-2) ◽  
pp. 59-62
Author(s):  
L. C. Biedenharnf ◽  
K. Srinivasa Rao
Keyword(s):  
Characteristic Feature ◽  
Quantum Group ◽  
Quantum Groups ◽  
Group Symmetry ◽  
Explicit Calculation ◽  
Matrix Elements ◽  
Structural Symmetry ◽  
Symmetry Properties ◽  
Q Factors ◽  
Tensor Operators

AbstractA characteristic feature of quantum groups is the occurrence of q-factors (factors of the form qk, k ∈ ℝ), which implement braiding symmetry. We show how the q-factors in matrix elements of elementary q-tensor operators (for all Uq(n)) may be evaluated, without explicit calculation, directly from structural symmetry properties.

scholarly journals A FINITE QUANTUM SYMMETRY OF $M(3, {\Bbb C})$

1998 ◽  
Vol 13 (24) ◽  
pp. 4147-4161 ◽  
Author(s):  
LUDWIK DABROWSKI ◽  
FABRIZIO NESTI ◽  
PASQUALE SINISCALCO
Keyword(s):  
Standard Model ◽  
Exact Sequence ◽  
Hopf Algebra ◽  
Quantum Group ◽  
Recent Work ◽  
Quantum Groups ◽  
Matrix Algebra ◽  
Group Symmetry ◽  
The Standard Model ◽  
The Matrix

The 27-dimensional Hopf algebra A(F), defined by the exact sequence of quantum groups [Formula: see text], [Formula: see text], is studied as a finite quantum group symmetry of the matrix algebra [Formula: see text], describing the color sector of Alain Connes' formulation of the Standard Model. The duality with the Hopf algebra ℋ, investigated in a recent work by Robert Coquereaux, is established and used to define a representation of ℋ on [Formula: see text] and two commuting representation of ℋ on A(F).

Symmetry properties of matrix elements of canonical SU(3) tensor operators

1994 ◽  
Vol 35 (12) ◽  
pp. 6672-6684 ◽  
Author(s):  
L. C. Biedenharn ◽  
M. A. Lohe ◽  
H. T. Williams
Keyword(s):  
Matrix Elements ◽  
Symmetry Properties ◽  
Tensor Operators

scholarly journals NONCOMMUTATIVITY IN A SIMPLE TOY MODEL

2007 ◽  
Vol 22 (14n15) ◽  
pp. 2643-2660 ◽  
Author(s):  
R. P. MALIK
Keyword(s):  
Phase Space ◽  
Gauge Symmetry ◽  
Quantum Group ◽  
Noncommutative Geometry ◽  
Quantum Groups ◽  
Present Model ◽  
Mass Parameter ◽  
Brst Symmetry ◽  
Symmetry Properties ◽  
Symmetry Transformations

We discuss various symmetry properties of the reparametrization invariant toy model of a free nonrelativistic particle and show that its commutativity and noncommutativity (NC) properties are the artifact of the underlying symmetry transformations. For the case of the symmetry transformations corresponding to the noncommutative geometry, the mass parameter of the toy model turns out to be noncommutative in nature. By exploiting the BRST symmetry transformations, we demonstrate the existence of this NC and show its cohomological equivalence with its commutative counterpart. A connection between the usual gauge symmetry transformations corresponding to the commutative geometry and the quantum groups, defined on the phase space, is also established for the present model at the level of Poisson bracket structure. We show that, for the noncommutative geometry, such a kind of quantum group connection does not exist.

q-Deformed Phase Space and its Lattice Structure

1997 ◽  
Vol 12 (28) ◽  
pp. 4997-5005 ◽  
Author(s):  
Julius Wess
Keyword(s):  
Phase Space ◽  
Euclidean Space ◽  
Quantum Group ◽  
Quantum Groups ◽  
Lattice Structure ◽  
Mathematical Structure ◽  
Group Symmetry ◽  
Configuration Spaces ◽  
Physical Systems ◽  
Noncommutative Spaces

Quantum groups lead to an algebraic structure that can be realized on quantum spaces. These are noncommutative spaces that inherit a well-defined mathematical structure from the quantum group symmetry. In turn such quantum spaces can be interpreted as noncommutative configuration spaces for physical systems. We study the noncommutative Euclidean space that is based on the quantum group SO q(3).

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 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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