q-Deformed orthogonal and pseudo-orthogonal algebras and their representations

Gavrilik, Alexander; Klimyk, A. U.

doi:10.1007/bf00420371

Cited by 84 publications

(94 citation statements)

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“…However, the Lie algebra so(n, C) has a different structure based on the basis elements I k,k−1 = E k,k−1 − E k−1,k (where E is is the matrix with elements (E is ) rt = δ ir δ st ). The q-deformation of this structure leads to the algebra U q (so(n, C)), determined in [1]. Later on it was shown that this q-deformation of so(n, C) is very useful in many directions of contemporary mathematics.…”

Section: Introductionmentioning

confidence: 98%

“…In this paper we consider most degenerate series representations of the q-deformed algebra so q (r, s), which is a real form of the complex q-deformed algebra U q (so(n, C)) defined in [1]. The algebra U q (so(n, C)) differs from the quantum algebra U q (so(n, C)) defined by Drinfeld [2] and Jimbo [3] (see also [4]).…”

Section: Introductionmentioning

confidence: 99%

“…This allows to construct Gel'fand-Tsetlin bases for finite dimensional representations of U q (so(n, C)) (see [1]). …”

Section: Introductionmentioning

confidence: 99%

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Degenerate Series Representations of the q-Deformed Algebra so'_q(r,s)

Groza¹

2007

SIGMA

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Abstract. The q-deformed algebra so q (r, s) is a real form of the q-deformed algebra U q (so(n, C)), n = r + s, which differs from the quantum algebra U q (so(n, C)) of Drinfeld and Jimbo. We study representations of the most degenerate series of the algebra so q (r, s). The formulas of action of operators of these representations upon the basis corresponding to restriction of representations onto the subalgebra so q (r) × so q (s) are given. Most of these representations are irreducible. Reducible representations appear under some conditions for the parameters determining the representations. All irreducible constituents which appear in reducible representations of the degenerate series are found. All * -representations of so q (r, s) are separated in the set of irreducible representations obtained in the paper.

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Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Degenerate Series Representations of the q-Deformed Algebra so'_q(r,s)

Groza¹

2007

SIGMA

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“…In [1] it was introduced a q-deformation of the universal enveloping algebra U (so n ) which uses a realization of the Lie algebra so n by skew-symmetric matrices (rather than by root elements and the diagonal Cartan subalgebra). If I ij = E ij − E ji are the skew-symmetric matrices of so n , where (E ij ) sr := δ is δ jr , then so n is generated by the matrices I 21 , I 32 , · · · , I n,n−1 (other basis matrices are obtained by taking commutators of these matrices).…”

Section: Introductionmentioning

confidence: 99%

“…The relations (4) and (5) restore the relations (2) and (3) if to introduce the element I 31 defined by (1). The algebra U ′ q (so 3 ) can be defined as the associative algebra generated by the elements I 21 and I 32 satisfying the defining relations (4) and (5).…”

Section: Introductionmentioning

confidence: 99%

Representations of the q-deformed algebra U_q'(SO₄)

Качурик

Klimyk

1994

J. Phys. A: Math. Gen.

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We study the nonstandard q-deformation U ′ q (so 4 ) of the universal enveloping algebra U (so 4 ) obtained by deforming the defining relations for skew-symmetric generators of U (so 4 ). This algebra is used in quantum gravity and algebraic topology. We construct a homomorphism φ of U ′ q (so 4 ) to the certain nontrivial extension of the Drinfeld-Jimbo quantum algebra⊗2 and show that this homomorphism is an isomorphism. By using this homomorphism we construct irreducible finite dimensional representations of the classical type and of the nonclassical type for the algebra U ′ q (so 4 ). It is proved that for q not a root of unity each irreducible finite dimensional representation of U ′ q (so 4 ) is equivalent to one of these representations. We prove that every finite dimensional representation of U ′ q (so 4 ) for q not a root of unity is completely reducible. It is shown how to construct (by using the homomorphism φ) tensor products of irreducible representations of U ′ q (so 4 ). (Note that no Hopf algebra structure is known for U ′ q (so 4 ).) These tensor products are decomposed into irreducible constituents.

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