# Explicit representations of classical Lie superalgebras in a Gelfand-Zetlin basis

N. I. Stoilova; J. Van der Jeugt

Banach Center Publications (2011)

- Volume: 93, Issue: 1, page 83-93
- ISSN: 0137-6934

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topN. I. Stoilova, and J. Van der Jeugt. "Explicit representations of classical Lie superalgebras in a Gelfand-Zetlin basis." Banach Center Publications 93.1 (2011): 83-93. <http://eudml.org/doc/282340>.

@article{N2011,

abstract = {An explicit construction of all finite-dimensional irreducible representations of classical Lie algebras is a solved problem and a Gelfand-Zetlin type basis is known. However the latter lacks the orthogonality property or does not consist of weight vectors for 𝔰𝔬(n) and 𝔰𝔭(2n). In case of Lie superalgebras all finite-dimensional irreducible representations are constructed explicitly only for 𝔤𝔩(1|n), 𝔤𝔩(2|2), 𝔬𝔰𝔭(3|2) and for the so called essentially typical representations of 𝔤𝔩(m|n). In the present paper we introduce an orthogonal basis of weight vectors for a class of infinite-dimensional representations of the orthosymplectic Lie superalgebra 𝔬𝔰𝔭(1|2n) and for all irreducible covariant tensor representations of the general linear Lie superalgebra 𝔤𝔩(m|n). Expressions for the transformation of the basis under the action of algebra generators are given. The results are a step towards the explicit construction of the parastatistics Fock space.},

author = {N. I. Stoilova, J. Van der Jeugt},

journal = {Banach Center Publications},

keywords = {Lie superalgebras; irreducible representations; matrix elements},

language = {eng},

number = {1},

pages = {83-93},

title = {Explicit representations of classical Lie superalgebras in a Gelfand-Zetlin basis},

url = {http://eudml.org/doc/282340},

volume = {93},

year = {2011},

}

TY - JOUR

AU - N. I. Stoilova

AU - J. Van der Jeugt

TI - Explicit representations of classical Lie superalgebras in a Gelfand-Zetlin basis

JO - Banach Center Publications

PY - 2011

VL - 93

IS - 1

SP - 83

EP - 93

AB - An explicit construction of all finite-dimensional irreducible representations of classical Lie algebras is a solved problem and a Gelfand-Zetlin type basis is known. However the latter lacks the orthogonality property or does not consist of weight vectors for 𝔰𝔬(n) and 𝔰𝔭(2n). In case of Lie superalgebras all finite-dimensional irreducible representations are constructed explicitly only for 𝔤𝔩(1|n), 𝔤𝔩(2|2), 𝔬𝔰𝔭(3|2) and for the so called essentially typical representations of 𝔤𝔩(m|n). In the present paper we introduce an orthogonal basis of weight vectors for a class of infinite-dimensional representations of the orthosymplectic Lie superalgebra 𝔬𝔰𝔭(1|2n) and for all irreducible covariant tensor representations of the general linear Lie superalgebra 𝔤𝔩(m|n). Expressions for the transformation of the basis under the action of algebra generators are given. The results are a step towards the explicit construction of the parastatistics Fock space.

LA - eng

KW - Lie superalgebras; irreducible representations; matrix elements

UR - http://eudml.org/doc/282340

ER -

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