Representation of the affine superalgebras \(A^{(4)}(0,2\ell)\), \(A^{(2)}(0,2\ell -1)\) and their subalgebras \(A_{2\ell}^{(2)}\), \(A^{(2)}_{2\ell -1}\) by vertex operators
type: Item
instance of: scholarly article
title: Representation of the affine superalgebras \(A^{(4)}(0,2\ell)\), \(A^{(2)}(0,2\ell -1)\) and their subalgebras \(A_{2\ell}^{(2)}\), \(A^{(2)}_{2\ell -1}\) by vertex operators
zbMATH Open document ID: 0652.17013
DOI: 10.1016/0021-8693(88)90250-5
author: George Golitzin
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: The Lepowsky-Wilson construction of representations of some twisted loop (Kac-Moody) algebras by vertex operators is generalized to those of twisted loop superalgebras that \[ \text{have a Cartan matrix and all of whose simple roots are nonisotropic.} \tag{*} \] (From the text one can not see that Kac-Moody algebras satisfying (*) constitute but a small portion of the whole list of them.) The generalization, whose straightforwardness is emphasized, still brings about some unexpected phenomena: the vacuum space with respect to the principal Heisenberg subalgebra is the tensor product of the ``usual'', nonsuper, vacuum by an exterior algebra. Therefore, the construction described is not as direct as, say, the description of irreducible finite-dimensional representations of osp(1/2n), the only finite- dimensional Lie superalgebra satisfying (*). Remark. Obviously, the generators of the above-mentioned exterior algebra and their dual with respect to the Killing form constitute a ``principal Heisenberg superalgebra'' which is interesting to compare with the one, isomorphic to gl(1/1), recently constructed by \textit{V. Kac} and \textit{J. van de Leur} [Ann. Inst. Fourier 37, No.4, 99-137 (1987; Zbl 0641.17011)].
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17A70
zbMATH DE Number: 4063329
zbMATH Keywords: affine superalgebras
zbMATH Keywords: standard modules of level one
zbMATH Keywords: highest weight
zbMATH Keywords: vertex operators
zbMATH Keywords: Kac-Moody algebras
reviewed by: Dimitry Leites
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90250-5
OpenAlex ID: W2023130539
cites work: Classical affine algebras
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cites work: Lie superalgebras
cites work: Infinite-dimensional algebras, Dedekind's \(\eta\)-function, classical Möbius function and the very strange formula
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cites work: The structure of standard modules. I: Universal algebras and the Rogers- Ramanujan identities
description: scientific article; zbMATH DE number 4063329