Representations of classical Lie algebras from their quantum deformations

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Abstract

We make use of a well-know deformation of the Poincaré Lie algebra in p + q + 1 dimensions (p + q > 0) to construct the Poincaré Lie algebra out of the Lie algebras of the de Sitter and anti de Sitter groups, the generators of the Poincaré Lie algebra appearing as certain irrational functions of the generators of the de Sitter groups. We have obtained generalizations of this "anti-deformation" for the SO(p + 2, q) and SO(p + 1, q + 1) cases with arbitrary p and q. Similar results have been established for q deformations Uq(so(p, q)) with small p and q values. Combining known results on representations of Uq(so(p, q)) (for q both generic and a root of unity) with our "anti-deformation" formulae, we get representations of classical Lie algebras which depend upon the deformation parameter q. Explicit results are given for the simplest example (of type A1) i.e. that associated with Uq(so(2, 1)).

Original languageEnglish (US)
Pages (from-to)683-686
Number of pages4
JournalInstitute of Physics Conference Series
Volume173
StatePublished - 2003

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algebra
generators
unity

All Science Journal Classification (ASJC) codes

  • Physics and Astronomy(all)

Cite this

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title = "Representations of classical Lie algebras from their quantum deformations",
abstract = "We make use of a well-know deformation of the Poincar{\'e} Lie algebra in p + q + 1 dimensions (p + q > 0) to construct the Poincar{\'e} Lie algebra out of the Lie algebras of the de Sitter and anti de Sitter groups, the generators of the Poincar{\'e} Lie algebra appearing as certain irrational functions of the generators of the de Sitter groups. We have obtained generalizations of this {"}anti-deformation{"} for the SO(p + 2, q) and SO(p + 1, q + 1) cases with arbitrary p and q. Similar results have been established for q deformations Uq(so(p, q)) with small p and q values. Combining known results on representations of Uq(so(p, q)) (for q both generic and a root of unity) with our {"}anti-deformation{"} formulae, we get representations of classical Lie algebras which depend upon the deformation parameter q. Explicit results are given for the simplest example (of type A1) i.e. that associated with Uq(so(2, 1)).",
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Representations of classical Lie algebras from their quantum deformations. / Moylan, Patrick J.

In: Institute of Physics Conference Series, Vol. 173, 2003, p. 683-686.

Research output: Contribution to journalArticle

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N2 - We make use of a well-know deformation of the Poincaré Lie algebra in p + q + 1 dimensions (p + q > 0) to construct the Poincaré Lie algebra out of the Lie algebras of the de Sitter and anti de Sitter groups, the generators of the Poincaré Lie algebra appearing as certain irrational functions of the generators of the de Sitter groups. We have obtained generalizations of this "anti-deformation" for the SO(p + 2, q) and SO(p + 1, q + 1) cases with arbitrary p and q. Similar results have been established for q deformations Uq(so(p, q)) with small p and q values. Combining known results on representations of Uq(so(p, q)) (for q both generic and a root of unity) with our "anti-deformation" formulae, we get representations of classical Lie algebras which depend upon the deformation parameter q. Explicit results are given for the simplest example (of type A1) i.e. that associated with Uq(so(2, 1)).

AB - We make use of a well-know deformation of the Poincaré Lie algebra in p + q + 1 dimensions (p + q > 0) to construct the Poincaré Lie algebra out of the Lie algebras of the de Sitter and anti de Sitter groups, the generators of the Poincaré Lie algebra appearing as certain irrational functions of the generators of the de Sitter groups. We have obtained generalizations of this "anti-deformation" for the SO(p + 2, q) and SO(p + 1, q + 1) cases with arbitrary p and q. Similar results have been established for q deformations Uq(so(p, q)) with small p and q values. Combining known results on representations of Uq(so(p, q)) (for q both generic and a root of unity) with our "anti-deformation" formulae, we get representations of classical Lie algebras which depend upon the deformation parameter q. Explicit results are given for the simplest example (of type A1) i.e. that associated with Uq(so(2, 1)).

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