Representations of the q-deformed algebra U<sub>q</sub>(iso<sub>2</sub>)
M Havlícek; A Klimyk; S Posta
Журнал:
Journal of Physics A: Mathematical and General
Дата:
1999-06-25
Аннотация:
An algebra homomorphism from the q-deformed algebra U<sub>q</sub>(iso<sub>2</sub>) with generating elements I, T<sub>1</sub>, T<sub>2</sub> and defining relations [I,T<sub>2</sub>]<sub>q</sub> = T<sub>1</sub>, [T<sub>1</sub>,I]<sub>q</sub> = T<sub>2</sub>, [T<sub>2</sub>,T<sub>1</sub>]<sub>q</sub> = 0 (where [A,B]<sub>q</sub> = q<sup>1/2</sup>AB-q<sup>-1/2</sup>BA) to the extension Û<sub>q</sub>(m<sub>2</sub>) of the Hopf algebra U<sub>q</sub>(m<sub>2</sub>) is constructed. The algebra U<sub>q</sub>(iso<sub>2</sub>) at q = 1 leads to the Lie algebra iso<sub>2</sub>~m<sub>2</sub> of the group ISO(2) of motions of the Euclidean plane. The Hopf algebra U<sub>q</sub>(m<sub>2</sub>) (which is not isomorphic to U<sub>q</sub>(iso<sub>2</sub>)) is treated as a Hopf q-deformation of the universal enveloping algebra of iso<sub>2</sub> and is well known in the literature. Not all irreducible representations of U<sub>q</sub>(m<sub>2</sub>) can be extended to representations of the extension Û<sub>q</sub>(m<sub>2</sub>). Composing the homomorphism with irreducible representations of Û<sub>q</sub>(m<sub>2</sub>) we obtain representations of U<sub>q</sub>(iso<sub>2</sub>). Not all of these representations of U<sub>q</sub>(iso<sub>2</sub>) are irreducible. The reducible representations of U<sub>q</sub>(iso<sub>2</sub>) are decomposed into irreducible components. In this way we obtain all irreducible representations of U<sub>q</sub>(iso<sub>2</sub>) when q is not a root of unity. A part of these representations turns into irreducible representations of the Lie algebra iso<sub>2</sub> when q1. Representations of the other part have no classical analogue.
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