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A New Family of Algebras Underlying the Rogers--Ramanujan Identities and Generalizations
James Lepowsky and Robert Lee Wilson
Proceedings of the National Academy of Sciences of the United States of America
Vol. 78, No. 12, [Part 1: Physical Sciences] (Dec., 1981), pp. 7254-7258
Published by: National Academy of Sciences
Stable URL: http://www.jstor.org/stable/11118
Page Count: 5
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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 A1 (1). Also, the present authors have introduced certain ``vertex'' differential operators providing a construction of A1 (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 LV which centralize the action of the principal Heisenberg subalgebra of an arbitrary Euclidean Lie algebra g on a highest weight g-module V. We state a general (tautological) Rogers--Ramanujan-type identity, which by our earlier theorem includes the classical identities, and we show that LV can be used to reformulate the general identity. For g=A1 (1), we develop the representation theory of LV 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 A1 (1), with an explicit realization of A1 (1) as operators in each case. The differential operator constructions mentioned above correspond to the trivial case LV = (1) of the present theory.
Proceedings of the National Academy of Sciences of the United States of America © 1981 National Academy of Sciences