Representations of theq-deformed algebraU_q(iso₂)

Abstract: An algebra homomorphism

p s i

from the q-deformed algebra

U_{q} ({m i s o}_{2})

with generating elements

I

,

T_{1}

,

T_{2}

and defining relations

[I, T_{2}]_{q} = T_{1}

,

[T_{1}, I]_{q} = T_{2}

,

[T_{2}, T_{1}]_{q} = 0

(where

[A, B]_{q} = q^{1 / 2} A B - q^{- 1 / 2} B A

) to the extension

{h a t U}_{q} ({m m}_{2})

of the Hopf algebra

U_{q} ({m m}_{2})

is constructed. The algebra

U_{q} ({m i s o}_{2})

at

q = 1

leads to the Lie algebra

{m i s o}_{2} s i m {m m}_{2}

of the group ISO(2) of motions of the Euclidean plane. The Hopf algebra

U_{q} ({m m}_{2})

is treated as a Hopf

q

-deformation of the universal enveloping algebra of

{m i s o}_{2}

and is well-known in the literature. Not all irreducible representations of

U_{q} ({m m}_{2})

can be extended to representations of the extension

{h a t U}_{q} ({m m}_{2})

. Composing the homomorphism

p s i

with irreducible representations of

{h a t U}_{q} ({m m}_{2})

we obtain representations of

U_{q} ({m i s o}_{2})

. Not all of these representations of

U_{q} ({m i s o}_{2})

are irreducible. The reducible representations of

U_{q} ({m i s o}_{2})

are decomposed into irreducible components. In this way we obtain all irreducible representations of

U_{q} ({m i s o}_{2})

when

q

is not a root of unity. A part of these representations turns into irreducible representations of the Lie algebra iso

_{2}

when

q o 1

. Representations of the other part have no classical analogue.

Representations of theq-deformed algebraUq(iso2)