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
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)].
scientific article; zbMATH DE number 4063329