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.

• Euclidean Kac-Moody Lie algebras
• standard modules
• principal Heisenberg subalgebras
• vacuum spaces