Standard modules for type one affine lie algebras
Standard modules for type one affine lie algebras
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一类仿射李代数的标准模块
DOI：
10.1007/bfb0071544
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发表时间：
1984
期刊：
影响因子：
--
通讯作者：
M. Primc
中科院分区：
文献类型：
--
作者：
J. Lepowsky;M. Primc
在[17]中，引入了~ V代数作为构造仿射Kac-Moody李代数的各种具体实现的程序的基础，这些李代数是由标准模标识的空间上的算子的李代数。[17]的详细处理[18]非常笼统，特殊情况包括从[5]、[20]开始研究的“均质图”，以及从[14]、[9]、[15]、[16]开始研究的“主图”。在本文中，我们对b[18]的均匀情况进行了更详细的研究。起始点是一类仿射李代数~，其中~是一个复单李代数；参见第2节。（关于方向、背景和参考资料，读者可以参考[5]和[13]。）利用与合适范畴中的x模V相关的代数~ V（第2节），我们用“广义交换关系”（第3节）定义的等价范畴重新表述了这些模的理论。然后我们构造了“广义Verma模块”(参见。[12])来使用这个视点（第4节）。到目前为止，这种材料是b[18]的特例。(本论文可以独立于b[18]阅读，尽管b[18]中有一些结构和论点写得更详细。读者可能会发现，目前的工作有助于激发b[18]的一般思想和结构。
In [17], algebras called~ V are introduced as the foundation of a program to construct a great variety of concrete realizations of the affine Kac-Moody Lie algebras, as Lie algebras of operators on spaces identified with the standard modules. The detailed treatment [18] of [17] is very general and includes as special cases the" homogeneous picture", whose study was begun in [5],[20], as well as the" principal picture", whose study was begun in [14],[9],[15],[16]. In this paper, we initiate the more detailed investigation of the homogeneous case of [18]. The starting point is a type one affine Lie algebra~, where~ is a complex simple Lie algebra; see Section 2.(For orientation, background and references, the reader can consult [5] and [13].) By means of the algebras~ V associated with the X-modules V in a suitable category (Section 2), we reformulate the theory of these modules in terms of an equivalent category defined by means of" generalized commutation relations"(Section 3). Then we construct the" generalized Verma modules"(cf.[12]) for~ using this viewpoint (Section 4). So far, this material is a special case of [18].(The present paper can be read independently of [18], although some constructions and arguments are written in a more detailed way in [18]. The reader might find that the present work helps to motivate the general ideas and constructions of [18].)