Explicit representations of classical Lie superalgebras in a Gelfand-Zetlin basis
Volume 93 / 2011
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 $\def\so{\mathfrak{so}}\so(n)$ 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.
