Quantum Group Symmetry and Q-Tensor Algebras

Biedenharn, L. C.; Lohe, M. A.

doi:10.1142/2815

Cited by 155 publications

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“…Next, we would like to turn our attention to some special representations of the symmetry generators L + , L 3 , and L − , namely spinor and vector representations [28,41]. The finite dimensional representations of U q (su 2 ) are already well-known (see for example Refs.…”

Section: Basic Ideas On Q-deformed Quantum Symmetriesmentioning

confidence: 99%

-Deformed quantum Lie algebras

Schmidt

Wachter

2006

Journal of Geometry and Physics

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Attention is focused on q-deformed quantum algebras with physical importance, i.e. U q (su 2 ), U q (so 4 ) and q-deformed Lorentz algebra. The main concern of this article is to assemble important ideas about these symmetry algebras in a consistent framework which shall serve as starting point for representation theoretic investigations in physics, especially quantum field theory. In each case considerations start from a realization of symmetry generators within the differential algebra. Formulae for coproducts and antipodes on symmetry generators are listed. The action of symmetry generators in terms of their Hopf structure is taken as q-analog of classical commutators and written out explicitly. Spinor and vector representations of symmetry generators are calculated. A review of the commutation relations between symmetry generators and components of a spinor or vector operator is given. Relations for the corresponding quantum Lie algebras are computed. Their Casimir operators are written down in a form similar to the undeformed case. *

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Section: Basic Ideas On Q-deformed Quantum Symmetriesmentioning

confidence: 99%

-Deformed quantum Lie algebras

Schmidt

Wachter

2006

Journal of Geometry and Physics

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“…Quantum groups are generalizations of the fundamental symmetry concepts of classical Lie groups and Lie algebras [9]. So quantum groups play an important role in carrying classical properties to the quantum structure.…”

Section: The Quantum Invariance Group Of Commuting Fermionsmentioning

confidence: 99%

The inhomogeneous quantum invariance group of commuting fermions

Altıntaş¹,

Arık²

2006

Open Physics

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“…It means that the addition of q-angular momenta depends on the order. So we have to deal with q-deformed Clebsh-Gordan Coefficients [11,12]:…”

Section: Q-deformed Angular Momentum Additionmentioning

confidence: 99%

“…There are many applications of q-algebras to all branches of physics. They are discussed in details in the books [11,5] and references therein.…”

Section: E Applications A) Rotational Spectra Of Pair-pair Nucleimentioning

confidence: 99%

Some unusual algebraic structures and their applications in many-body problems

Avancini¹

2003

Braz. J. Phys.

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In this paper, we introduce the q-deformed and quon algebras formalism. Some applications of these algebraic structures are considered. A possible connection of the quon algebra with composite particle systems is discussed and perspectives on using such mathematical objects in the Bose-Einstein condensation is presented. I IntroductionThe identification of algebraic structures in quantum physical systems has been an important tool for their understanding. Concepts of symmetry, invariance and group theory have shown to be of great utility from the beginning of quantum mechanics [1]. In a more recent context some unusual algebraic structures related to quantum inverse scattering methods and statistical mechanics models have appeared [2]. These algebraic structures are usually called q-algebras, qdeformed algebras or quantum groups. In current theoretical investigations there have been extensive applications of qalgebras to many branches of physics [3,4,5]. In a sense, many successful standard applications of group theory in the past may be extended to a quantum group symmetry. The aim of this paper is to discuss the use of q-algebras in the framework of many-body problems. We introduce the mathematical fundamentals as simply as possible and consider applications to q-deformed pseudo-spin models. These models can be considered as convenient theoretical laboratories where one can test the properties of these algebraic structures. We also show that the q-algebra formalism may be useful in the context of boson mappings,i. e., when the substitution of fermion pairs by bosons is convenient. A q-deformed algebra has a closely related algebraic structure which is called the quon algebra [6,7]. Such algebra was proposed to describe particles that violate statistics by a small amount. We consider the applications of quon algebras in boson mappings and nuclear models. We also show that they may be relevant in the study of composite bosonic particles (particles composed by fermion pairs), since deviations of a true bosonic particle could be accommodated in a natural way through the quon algebra. This formalism can also be applied to the study of Bose-Einstein condensation in trapped atoms [8]. This paper is organized as follows: In Sec.II quantum groups or q-algebras are defined and some of their properties and applications are analysed, in Sec.III the quon algebra formalism is considered and ,finally, in Sec.IV we present our conclusions. II q-deformed algebras or quantum groupsThe q-algebras or quantum groups are generalizations of classical Lie groups and Lie algebras and involve two fundamental ideas: Deformation and Non-commutative comultiplication [5]. Next, these two concepts will be discussed in detail. A. q-deformed objectsA q-boson algebra [9,10] (deformed harmonic oscillator) is a set of elements called q-boson operators: a (annihilation), a † (creation), N (number), which satisfy the following commutation relations:Note that the first and second commutation relations are equal to the common harmonic oscillator, ho...

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Quantum Group Symmetry and Q-Tensor Algebras

Cited by 155 publications

References 0 publications

-Deformed quantum Lie algebras

-Deformed quantum Lie algebras

The inhomogeneous quantum invariance group of commuting fermions

Some unusual algebraic structures and their applications in many-body problems

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