文章基本信息

标题：The Rogers-Ramanujan identities: Lie theoretic interpretation and proof
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作者：James Lepowsky ; Robert Lee Wilson
期刊名称：Proceedings of the National Academy of Sciences
印刷版ISSN：0027-8424
电子版ISSN：1091-6490
出版年度：1981
卷号：78
期号：2
页码：699-701
DOI：10.1073/pnas.78.2.699
语种：English
出版社：The National Academy of Sciences of the United States of America
摘要：The two Rogers-Ramanujan identities, which equate certain infinite products with infinite sums, are among the most intriguing of the classical formal power series identitites. It has been found by Lepowsky and Milne that the product side of each of them differs by a certain factor from the principally specialized character of a certain standard module for the Euclidean Kac-Moody Lie algebra A1(1). On the other hand, the present authors have introduced an infinite-dimensional Heisenberg subalgebra [unk] of A1(1) which leads to a construction of A1(1) in terms of differential operators given by the homogeneous components of an "exponential generating function." In the present announcement, we use [unk] to formulate a natural "abstract Rogers-Ramanujan identity" for an arbitrary standard A1(1)-module which turns out to coincide with the classical identities in the cases of the two corresponding standard modules. The abstract identity equates two expressions, one a product and the other a sum, for the principally specialized character of the space {Omega} of highest weight vectors or "vacuum states" for [unk] in the module. The construction of A1(1) leads to a concrete realization of {Omega} as the span of certain spaces of symmetric polynomials occurring as the homogeneous components of exponential generating functions. The summands in the Rogers-Ramanujan identities turn out to "count" the dimensions of these spaces. For general standard A1(1)-modules, we conjecture that the abstract identities agree with generalizations of the Rogers-Ramanujan identities due to Gordon, Andrews, and Bressoud.
关键词：Euclidean Kac-Moody Lie algebras ; standard modules ; principally specialized characters ; infinite-dimensional Heisenberg algebras