Basic Representations for Classical Affine Lie Algebras

AbstractPresented 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+g0+g1 of a classical Lie algebra g and the corresponding decomposition g̃=g̃−1+g̃0+g̃1 of the affine Lie algebra g̃. By using a generalization of the Frenkel–Kac vertex operator formula for A(1)1 one can construct a spanning set of the basic g̃-module in terms of monomials in basis elements of g̃1 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 g0-module g1. The linear independence of the constructed spanning set of the basic g̃-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