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

Volume 93 / 2011

N. I. Stoilova, J. Van der Jeugt Banach Center Publications 93 (2011), 83-93 MSC: Primary 17Bxx; Secondary 17B81. DOI: 10.4064/bc93-0-7

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 and \def\sp{\mathfrak{sp}}\sp(2n). In case of Lie superalgebras all finite-dimensional irreducible representations are constructed explicitly only for \def\gl{\mathfrak{gl}}\gl(1|n), \def\gl{\mathfrak{gl}}\gl(2|2), \def\osp{\mathfrak{osp}}\osp(3|2) and for the so called essentially typical representations of \def\gl{\mathfrak{gl}}\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 \def\osp{\mathfrak{osp}}\osp(1|2n) and for all irreducible covariant tensor representations of the general linear Lie superalgebra \def\gl{\mathfrak{gl}}\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.

Authors

  • N. I. StoilovaDepartment of Applied Mathematics and Computer Science
    Ghent University
    Krijgslaan 281-S9
    B-9000 Gent, Belgium
    e-mail
  • J. Van der JeugtDepartment of Applied Mathematics and Computer Science
    Ghent University
    Krijgslaan 281-S9
    B-9000 Gent, Belgium
    e-mail

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