A new family of algebras underlying the Rogers-Ramanujan identities and generalizations

Abstract

The classical Rogers-Ramanujan identities have been interpreted by Lepowsky-Milne and the present authors in terms of the representation theory of the Euclidean Kac-Moody Lie algebra A(1) ((1)). Also, the present authors have introduced certain "vertex" differential operators providing a construction of A(1) ((1)) on its basic module, and Kac, Kazhdan, and we have generalized this construction to a general class of Euclidean Lie algebras. Starting from this viewpoint, we now introduce certain new algebras [unk](v) which centralize the action of the principal Heisenberg subalgebra of an arbitrary Euclidean Lie algebra [unk] on a highest weight [unk]-module V. We state a general (tautological) Rogers-Ramanujan-type identity, which by our earlier theorem includes the classical identities, and we show that [unk](v) can be used to reformulate the general identity. For [unk] = A(1) ((1)), we develop the representation theory of [unk](v) in considerable detail, allowing us to prove our earlier conjecture that our general Rogers-Ramanujan-type identity includes certain identities of Gordon, Andrews, and Bressoud. In the process, we construct explicit bases of all of the standard and Verma modules of nonzero level for A(1) ((1)), with an explicit realization of A(1) ((1)) as operators in each case. The differential operator constructions mentioned above correspond to the trivial case [unk](v) = (1) of the present theory.