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

N. 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 -