有限单李伪代数上的不可约模 III. H型本原伪代数
Irreducible modules over finite simple Lie pseudoalgebras III. Primitive pseudoalgebras of type H
AI总结：
本文是单李伪代数表示论系列第三篇，证明了H型本原李伪代数的有限不可约模结构定理，并引入共形辛伪de Rham复形刻画了其不可约模分类。
摘要：
李共形代数是一种代数结构，它编码了共形场论中手性场的算子积展开的奇异部分。李伪代数是这一结构的推广，其中未定元中的多项式代数 k[∂] 被基域 k 上有限维李代数 d 的泛包络代数 U(d) 所替代。有限（即在 U(d) 上有限生成）单李伪代数已在我们的2001年论文 [BDK] 中被分类。完整列表由 W、S、H 和 K 型本原李伪代数，以及它们或单有限维李代数上的流李伪代数组成。本文是我们关于单李伪代数表示论系列论文的第三篇。在第一篇论文中，我们证明了 W 或 S 型本原李伪代数上的任何有限不可约模，要么是不可约张量模，要么是伪 de Rham 复形某一成员中微分的像。在第二篇论文中，我们对 K 型本原李伪代数建立了一个类似的结果，其中伪 de Rham 复形被一个特定的约化所替代，称为切触伪 de Rham 复形。这种在切触几何背景下的约化是由 M. Rumin [Rum] 发现的。在本文中，我们证明对于 H 型本原李伪代数，一个类似于 K 型的结果成立，其中切触伪 de Rham 复形被一个合适的复形所替代。然而，H 型的情况更为复杂，因为湮灭代数不像其他情况那样是对应的李-Cartan 代数，而是一个不可约中心扩张。当湮灭代数中心的作用平凡时，该复形与 M. Eastwood [E] 在共形辛几何上的工作相关，我们将其称为共形辛伪 de Rham 复形。
英文摘要：
A Lie conformal algebra is an algebraic structure that encodes the singular part of the operator product expansion of chiral fields in conformal field theory. A Lie pseudoalgebra is a generalization of this structure, for which the algebra of polynomials k[\partial] in the indeterminate is replaced by the universal enveloping algebra U(d) of a finite-dimensional Lie algebra d over the base field k. The finite (i.e., finitely generated over U(d)) simple Lie pseudoalgebras were classified in our 2001 paper [BDK]. The complete list consists of primitive Lie pseudoalgebras of type W, S, H, and K, and of current Lie pseudoalgebras over them or over simple finite-dimensional Lie algebras. The present paper is the third in our series on representation theory of simple Lie pseudoalgebras. In the first paper, we showed that any finite irreducible module over a primitive Lie pseudoalgebra of type W or S is either an irreducible tensor module or the image of the differential in a member of the pseudo de Rham complex. In the second paper, we established a similar result for primitive Lie pseudoalgebras of type K, with the pseudo de Rham complex replaced by a certain reduction, called the contact pseudo de Rham complex. This reduction in the context of contact geometry was discovered by M. Rumin [Rum]. In the present paper, we show that for primitive Lie pseudoalgebras of type H, a similar to type K result holds with the contact pseudo de Rham complex replaced by a suitable complex. However, the type H case in more involved, since the annihilation algebra is not the corresponding Lie-Cartan algebra, as in other cases, but an irreducible central extension. When the action of the center of the annihilation algebra is trivial, this complex is related to work by M. Eastwood [E] on conformally symplectic geometry, and we call it conformally symplectic pseudo de Rham complex.
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