Basic Representations for Classical Affine Lie Algebras,

Presented here is a construction of certain bases of basic representations for classical affine Lie algebras. The starting point is a -grading = − 1 + 0 + 1 of a classical Lie algebra and the corresponding decomposition = − 1 + 0 + 1 of the affine Lie algebra . By using a generalization of the Frenkel–Kac vertex operator formula for A(1)1 one can construct a spanning set of the basic -module in terms of monomials in basis elements of 1 and certain group element e. These monomials satisfy certain combinatorial Rogers–Ramanujan type difference conditions arising from the vertex operator formula, and the main result is that these differences coincide with the energy function of a perfect crystal corresponding to the 0-module 1. The linear independence of the constructed spanning set of the basic -module is proved by using a crystal base character formula for standard modules due to S.-J. Kang, M. Kashiwara, K. C. Misra, T. Miwa, T. Nakashima, and A. Nakayashiki.