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('). Also, the present authors have introduced certain "vertex" differential operators providing a construction ofA(') 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 2v which centralize the action ofthe principal Heisenberg subalgebra of an arbitrary Euclidean Lie algebra g on a highest weight gmodule V. We state a general (tautological) Rogers-Ramanujantype identity, which by our earlier theorem includes the classical identities, and we show that 2v can be used to reformulate the general identity. For g = A('), we develop the representation theory of 2v 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('), with an explicit realization of A() as operators in each case. The differential operator constructions mentioned above correspond to the trivial case 2v = (1) of the present theory.In this paper, we launch a program to give explicit constructions of general standard modules of general Euclidean Lie algebras and, hence, to produce a wide variety of new realizations of these Lie algebras as algebras of operators. The first construction (1) of a Euclidean Lie algebra, namely A('), by differential operators on a "Fock space" and its sequel (2) for a general class of Euclidean Lie algebras turn out to be the "trivial" cases of the present theory, in a sense to be made precise below. The modules V under consideration may each be viewed as the tensor product of a "Fock space" with a "vacuum space" Ql, for the principal Heisenberg subalgebra ? of g. Our algebraThe publication costs ofthis article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U. S. C. §1734 solely to indicate this fact.1v commutes with the action of B and, hence, preserves [1 and acts richly enough on it essentially to "untwist" the action of g to the tensor product of two commuting actions. A striking feature of1v is that, in its action on V, it satisfies identities that are themselves the "generating functions" of infinite systems of identities. In this paper, these identities are

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