dorsal/arxiv
View SchemaA nonlinear deformed su(2) algebra with a two-colour quasitriangular Hopf structure
| Authors | D. Bonatsos, C. Daskaloyannis, P. Kolokotronis, A. Ludu, C. Quesne |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9701029 |
| URL | https://arxiv.org/abs/q-alg/9701029 |
| DOI | 10.1063/1.531854 |
| Journal | J. Math. Phys. 38 (1997) 369-386 |
Abstract
Nonlinear deformations of the enveloping algebra of su(2), involving two arbitrary functions of J_0 and generalizing the Witten algebra, were introduced some time ago by Delbecq and Quesne. In the present paper, the problem of endowing some of them with a Hopf algebraic structure is addressed by studying in detail a specific example, referred to as ${\cal A}^+_q(1)$. This algebra is shown to possess two series of (N+1)-dimensional unitary irreducible representations, where N=0, 1, 2, .... To allow the coupling of any two such representations, a generalization of the standard Hopf axioms is proposed by proceeding in two steps. In the first one, a variant and extension of the deforming functional technique is introduced: variant because a map between two deformed algebras, su_q(2) and ${\cal A}^+_q(1)$, is considered instead of a map between a Lie algebra and a deformed one, and extension because use is made of a two-valued functional, whose inverse is singular. As a result, the Hopf structure of su_q(2) is carried over to ${\cal A}^+_q(1)$, thereby endowing the latter with a double Hopf structure. In the second step, the definition of the coproduct, counit, antipode, and R-matrix is extended so that the double Hopf algebra is enlarged into a new algebraic structure. The latter is referred to as a two-colour quasitriangular Hopf algebra because the corresponding R-matrix is a solution of the coloured Yang-Baxter equation, where the `colour' parameters take two discrete values associated with the two series of finite-dimensional representations.
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"abstract": "Nonlinear deformations of the enveloping algebra of su(2), involving two\narbitrary functions of J_0 and generalizing the Witten algebra, were introduced\nsome time ago by Delbecq and Quesne. In the present paper, the problem of\nendowing some of them with a Hopf algebraic structure is addressed by studying\nin detail a specific example, referred to as ${\\cal A}^+_q(1)$. This algebra is\nshown to possess two series of (N+1)-dimensional unitary irreducible\nrepresentations, where N=0, 1, 2, .... To allow the coupling of any two such\nrepresentations, a generalization of the standard Hopf axioms is proposed by\nproceeding in two steps. In the first one, a variant and extension of the\ndeforming functional technique is introduced: variant because a map between two\ndeformed algebras, su_q(2) and ${\\cal A}^+_q(1)$, is considered instead of a\nmap between a Lie algebra and a deformed one, and extension because use is made\nof a two-valued functional, whose inverse is singular. As a result, the Hopf\nstructure of su_q(2) is carried over to ${\\cal A}^+_q(1)$, thereby endowing the\nlatter with a double Hopf structure. In the second step, the definition of the\ncoproduct, counit, antipode, and R-matrix is extended so that the double Hopf\nalgebra is enlarged into a new algebraic structure. The latter is referred to\nas a two-colour quasitriangular Hopf algebra because the corresponding R-matrix\nis a solution of the coloured Yang-Baxter equation, where the `colour\u0027\nparameters take two discrete values associated with the two series of\nfinite-dimensional representations.",
"arxiv_id": "q-alg/9701029",
"authors": [
"D. Bonatsos",
"C. Daskaloyannis",
"P. Kolokotronis",
"A. Ludu",
"C. Quesne"
],
"categories": [
"q-alg",
"hep-th",
"math-ph",
"math.MP",
"math.QA"
],
"doi": "10.1063/1.531854",
"journal_ref": "J. Math. Phys. 38 (1997) 369-386",
"title": "A nonlinear deformed su(2) algebra with a two-colour quasitriangular Hopf structure",
"url": "https://arxiv.org/abs/q-alg/9701029"
},
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