Explicit Representations of Classical Lie Superalgebras in a Gelf

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 for the explicit construction of the parastatistics Fock space.