Approximate symmetries and quantum algebras

Celeghini, E.; Olmo, M. A. del

doi:10.1209/epl/i2003-00326-y

Cited by 3 publications

(6 citation statements)

References 8 publications

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“…This symmetry property of the coproduct can be imposed in any dimension and makes much easier the procedures of symmetrization and 'hermitation' [13]. In particular, given a certain Lie bialgebra (a, η), the above assumption implies that the first order deformation of the coproduct will be just given by the (skewsymmetric) Lie bialgebra cocommutator.…”

Section: Generalized Cocommutativitymentioning

confidence: 99%

“…We point out that the definition (1.1) for the underlying cocommutator implies that the deformation parameter appear explicitly as multiplicative factors within the cocommutator. This symmetry property of the coproduct can be imposed in any dimension and makes much easier the procedures of symmetrization and 'hermitation' [13]. In particular, given a certain Lie bialgebra (a, η), the above assumption implies that the first order deformation of the coproduct will be just given by the (skewsymmetric) Lie bialgebra cocommutator.…”

Section: Generalized Cocommutativitymentioning

confidence: 99%

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Three-dimensional quantum algebras: a Cartan-like point of view

Ballesteros¹,

Celeghini²,

Olmo³

2004

J. Phys. A: Math. Gen.

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A perturbative quantization procedure for Lie bialgebras is introduced and used to classify all three dimensional complex quantum algebras compatible with a given coproduct. The role of elements of the quantum universal enveloping algebra that, analogously to generators in Lie algebras, have a distinguished type of coproduct is discussed, and the relevance of a symmetrical basis in the universal enveloping algebra stressed. New quantizations of three dimensional solvable algebras, relevant for possible physical applications for their simplicity, are obtained and all already known related results recovered. Our results give a quantization of all existing three dimensional Lie algebras and reproduce, in the classical limit, the most relevant sector of the complete classification for real three dimensional Lie bialgebra structures given in [1].MSC: 81R50, 81R40, 17B37

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Section: Generalized Cocommutativitymentioning

confidence: 99%

Section: Generalized Cocommutativitymentioning

confidence: 99%

Three-dimensional quantum algebras: a Cartan-like point of view

Ballesteros¹,

Celeghini²,

Olmo³

2004

J. Phys. A: Math. Gen.

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“…Besides the proposed almost primitive basis, we would like to quote two other relevant bases that play a role both in mathematics and in physics: the Lie basis and the canonical/crystal basis. In the Lie basis (for instance, in twisting) the algebra remains unmodified and all deformations affect the coalgebra offering a possible way to introduce an interaction but saving the global symmetry [15]. In the canonical or crystal basis (with applications in statistical mechanics [16] and in genetics [17])), instead, the algebraic sector of the Hopf algebra structure is obtained in the limit |z| → ∞ [18,19] and coalgebra is a byproduct.…”

Section: Discussionmentioning

confidence: 99%

From quantum universal enveloping algebras to quantum algebras

Celeghini

Ballesteros

Olmo

2008

J. Phys. A: Math. Theor.

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The "local" structure of a quantum group G q is currently considered to be an infinite-dimensional object: the corresponding quantum universal enveloping algebra U q (g), which is a Hopf algebra deformation of the universal enveloping algebra of a n-dimensional Lie algebra g = Lie(G). However, we show how, by starting from the generators of the underlying Lie bialgebra (g, δ), the analyticity in the deformation parameter(s) allows us to determine in a unique way a set of n "almost primitive" basic objects in U q (g), that could be properly called the "quantum algebra generators". So, the analytical prolongation (g q , ∆) of the Lie bialgebra (g, δ) is proposed as the appropriate local structure of G q . Besides, as in this way (g, δ) and U q (g) are shown to be in one-to-one correspondence, the classification of quantum groups is reduced to the classification of Lie bialgebras. The su q (2) and su q (3) cases are explicitly elaborated.MSC: 81R50, 16W30, 17B37

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“…Note also that a choice among the bases has been informally done by the scientific community: without any official justification almost always U q (su(2)) is written in a basis similar to that described in eqs. (7,8), related to the analyticity of the whole scheme. Analyticity indeed allows to select this basis, we call analytical basis.…”

Section: Introductionmentioning

confidence: 99%

“…The co-product, on the contrary, is determined by the coalgebra to be not primitive. This basis could describe a conserved Lie symmetry where the presence of z = 0 is a signal of one interaction in the composed system [8]. 4) a basis g ∞ obtained in the limit z ′ → ∞, i.e.…”

Section: Introductionmentioning

confidence: 99%

Quantum Bases in Uq(g)

Celeghini

2008

Preprint

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This paper is devoted to analyze, inside the ∞-many possible bases of a Quantum Universal Enveloping Algebra U q (g), those that can be considered as "more equal then others", like orthonormal bases in the Euclidean spaces.The only possible element of selection has been found to be a privileged connection with the corresponding bialgebra. A new parameter z ′ ∈ C -independent from the z := log(q) ∈ C that defines U q (g)-has thus been introduced. Each value of such new parameter z ′ defines one of these bases (we call quantum bases) and determines, independently from z, its commutation relations. Both z and z ′ are, on the contrary, necessary to fix the relations between the basic set and its co-products. We have thus -for each pair {z, z ′ }-one n-dimensional quantum basis (g z ′ , ∆ zz ′ ), that describes U q (g).Three cases are particularly relevant: the analytical basis g z , where z ′ = z, the Lie basis g 0 obtained for z ′ = 0 (where both the basic set and its co-product close Lie-like commutation relations and the non primitive co-product describe an interaction) and the canonical/crystal basis g ∞ , limit for z ′ → ∞ in the Riemann plane of the generic quantum basis.To simplify the exposition, we discuss in details the easily generalizable case of U q (su(2)).

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Approximate symmetries and quantum algebras

Cited by 3 publications

References 8 publications

Three-dimensional quantum algebras: a Cartan-like point of view

Three-dimensional quantum algebras: a Cartan-like point of view

From quantum universal enveloping algebras to quantum algebras

Quantum Bases in Uq(g)

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