School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540.
Proc Natl Acad Sci U S A. 1981 Dec;78(12):7254-8. doi: 10.1073/pnas.78.12.7254.
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 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 can be used to reformulate the general identity. For [unk] = A(1) ((1)), we develop the representation theory of unk 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 = (1) of the present theory.
经典的 Rogers-Ramanujan 恒等式已经被 Lepowsky-Milne 和本作者从欧几里得 Kac-Moody Lie 代数 A(1) ((1)) 的表示论的角度进行了解释。此外，本作者还引入了某些“顶点”微分算子，提供了在其基本模上构造 A(1) ((1)) 的方法，并且 Kac、Kazhdan 和我们已经将这种构造推广到了一般的欧几里得 Lie 代数类。从这个角度出发，我们现在引入了某些新的代数 unk，它们中心化了任意欧几里得 Lie 代数 [unk] 的主 Heisenberg 子代数在最高权 [unk]-模 V 上的作用。我们陈述了一个一般的（ tautological）Rogers-Ramanujan 型恒等式，通过我们之前的定理，这个恒等式包含了经典的恒等式，并且我们表明 unk 可以用于重新表述这个一般的恒等式。对于 [unk] = A(1) ((1))，我们详细地发展了 unk 的表示论，这使我们能够证明我们之前的猜想，即我们的一般 Rogers-Ramanujan 型恒等式包含了 Gordon、Andrews 和 Bressoud 的某些恒等式。在这个过程中，我们构建了 A(1) ((1)) 的所有非零级别的标准和 Verma 模的显式基，并在每种情况下都显式地实现了 A(1) ((1)) 作为算子。上述微分算子构造对应于本理论的平凡情况 unk = (1)。
