Finite-dimensional representations of the quantum superalgebraU q[gl(n/m)] and relatedq-identities

Palev, T. D.; Stoilova, N. I.; Jeugt, J. Van der

doi:10.1007/bf02112320

Cited by 37 publications

(35 citation statements)

References 33 publications

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“…The representation theory of these algebras have been much studied (see, e.g., [1,13,15,17]), and their applications to the theory of knots and 3-manifolds led to the discovery of many new topological invariants. The finite-dimensional unitary representations of the quantum general linear superalgebra were classified in [5].…”

Section: Introductionmentioning

confidence: 99%

Unitary highest weight representations of quantum general linear superalgebra

Zhang

2009

Journal of Algebra

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Fock space realisations of unitary highest weight representations of compact and noncompact real forms of the quantum general linear superalgebra U q (gl m|n ) are constructed. We decompose all tensor powers of the Fock spaces into direct sums of irreducible unitary submodules for compact or noncompact real forms of the quantum superalgebra, obtaining multiplicities of the irreducibles and also explicit formulae for their highest weight vectors. A generalised Howe duality of type (U q (gl m|n ), U q (gl k )) is also established, which is used in the study of unitary representations and should also be of independent interest.

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Section: Introductionmentioning

confidence: 99%

Unitary highest weight representations of quantum general linear superalgebra

Zhang

2009

Journal of Algebra

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“…(For the non-standard two-parameter deformations of GL(1=1) we refer to [12], [13], [14].) In the case of oneparametric deformation the superalgebra U q (gl(m=n)) in duality with GL q (m=n) and its quantum subsuperalgebra U q (sl(m=n)) were studied in, e.g., [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35]. However, there was not much study of the multiparameter deformations of U(gl(m=n)) and U(sl(m=n)) and their interrelations, namely, t w o-parameter deformations were obtained for m = n = 1 in [36], [5], [8], and multiparameter deformations of U(sl(m=n)) were obtained in [37], and of U(sl(m=1)) in [38].…”

Section: Introductionmentioning

confidence: 99%

Induced representations of the multiparameter Hopf Superalgebras U_uq(gl(m/n)) and U_uq(sl(m/n))

Dobrev

Tahri

1999

J. Phys. A: Math. Gen.

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We construct induced representations of the multiparameter Hopf superalgebras U uq (gl(m=n)) and U uq (sl(m=n)). The rst superalgebra we constructed earlier as the dual of the multiparameter quantum deformation of the supergroup GL(m=n). The second superalgebra is a Hopf subalgebra of the rst for a special choice of the parameters. The representations are labelled by m + n integer numbers, respectively m + n 1 complex numbers, and act in the space of formal power series of (m + n)(m + n 1)=2 noncommuting variables, of which mn are odd and the rest are even. These variables generate a q-deformation of a ag supermanifold of the supergroup GL(m=n), respectively SL(m=n).2

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“…Irreducible representations of the quantum analogue of superalgebras are studied intensively when q is not root of unity in [20,21,18,25].…”

Section: Introductionmentioning

confidence: 99%

Centre and representations of at roots of unity

Abdesselam¹,

Arnaudon²,

Bauer³

1997

J. Phys. A: Math. Gen.

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Quantum groups at roots of unity have the property that their centre is enlarged. Polynomial equations relate the standard deformed Casimir operators and the new central elements. These relations are important from a physical point of view since they correspond to relations among quantum expectation values of observables that have to be satisfied on all physical states. In this paper, we establish these relations in the case of the quantum Lie superalgebra U q (sl(2|1)). In the course of the argument, we find and use a set of representations such that any relation satisfied on all the representations of the set is true in U q (sl(2|1)). This set is a subset of the set of all the finite dimensional irreducible representations of U q (sl(2|1)), that we classify and describe explicitly.

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Finite-dimensional representations of the quantum superalgebraU q[gl(n/m)] and relatedq-identities

Cited by 37 publications

References 33 publications

Unitary highest weight representations of quantum general linear superalgebra

Unitary highest weight representations of quantum general linear superalgebra

Induced representations of the multiparameter Hopf Superalgebras U_uq(gl(m/n)) and U_uq(sl(m/n))

Centre and representations of at roots of unity

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Finite-dimensional representations of the quantum superalgebraU q[gl(n/m)] and relatedq-identities

Cited by 37 publications

References 33 publications

Unitary highest weight representations of quantum general linear superalgebra

Unitary highest weight representations of quantum general linear superalgebra

Induced representations of the multiparameter Hopf Superalgebras Uuq(gl(m/n)) and Uuq(sl(m/n))

Centre and representations of at roots of unity

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Induced representations of the multiparameter Hopf Superalgebras U_uq(gl(m/n)) and U_uq(sl(m/n))