Representation of the affine superalgebras A(4)(0, 2l), A(2)(0, 2l - 1) and their subalgebras A2l(2), A2l - 1(2) by vertex operators

Abstract

The structure theory of standard modules of affine Lie algebras, given by J. Lepowsky and R. L. Wilson in [LW], is stated for representations of affine superalgebras. As an application, the standard modules of level one for the superalgebras A(4)(0, 2l), A(2)(0, 2l - 1) and their affine subalgebras A2l(2), A2l - 1(2) are constructed explicitly. These modules are realized as the tensor product of symmetric and exterior algebras with an irreducible representation of a certain finite 2-group. The affine superalgebra acts on this space by tensor products of vertex operators, operators of Clifford type, and elements of the 2-group. As a corollary, the spin representations of the Lie algebras Bl, and Dl are obtained from the 2-group representation.