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 Z-grading g = g(-1) + g(0) + g(1) of a classical Lie algebra g and the corresponding decomposition (g) over tilde = (g) over tilde(-1) + (g) over tilde(0) + (g) over tilde(1) of the affine Lie algebra g. By using a generalization of the Frenkel-Kac vertex operator formula for A(1)((1)) one can construct a spanning set of the basic (g) over tilde-module in terms of monomials in basis elements of (g) over tilde(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 g(0)-module g(1). The linear independence of the constructed spanning set of the basic (g) over tilde-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. (C) 2000 Academic Press.