Derivations and central extensions of finitely generated graded Lie algebras
Derivations and central extensions of finitely generated graded Lie algebras
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DOI：
10.1016/0021-8693(88)90046-4
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发表时间：
1988-10
影响因子：
0.9
通讯作者：
Rolf Farnsteiner
中科院分区：
文献类型：
--
作者：
Rolf Farnsteiner
李论的推导的意义主要在于它们对低维上同群的亲和力。因此，他们的决定经常提供洞察李代数的结构特征，而这些特征在定义性质中并不突出。在同构问题中，导数的向量空间Der，（L， L）和对偶空间导数的向量空间Der，（L, L*）是特别有趣的。在后一种情况下，H ‘ （L, L*）和H ’ （L: F）之间的密切相互关系表明，可以利用衍生来对李代数的中心扩展进行分类（参见[5,61]）。研究有限维李代数最有效的工具之一是相对于Cartan子代数的根空间分解。Block [IS]在确定经典和一些非经典简单李代数的中心扩展时，广泛使用了基于相应层次的技术。最近，Winter b[13]使用类似的方法来研究经典Albert-Zassenhaus代数的推导。这些方法甚至对无限维代数也能产生结果，这一事实是伯曼首先发现的[3:41]。本文研究了在梯度模中有值的梯度李代数的导数的确定。我们的结果适用于任意维的有限生成李代数，为上述论文提供了一个概念框架。我们将33岁
The significance of derivations for Lie theory primarily resides in their affinity to low dimensional cohomology-groups. Their determination therefore frequently affords insight into structural features of Lie algebras which do not figure prominently in the defining properties. In connection with isomorphism problems the vector spaces Der,(L, L) and Der,(L, L*) of derivations and dual space derivations, respectively, are of particular interest. In the latter case the close interrelation between H’(L, L*) and H’(L: F) reveals that derivations may be utilized in order to classify central extensions of Lie algebras (cf.[5, 61). One of the most efficient tools in the investigation of finite dimensional Lie algebras is the root space decomposition relative to a Cartan subalgebra. Techniques based on the corresponding gradation were extensively employed by Block in [IS] in the determination of the central extensions of the classical and some non-classical simple Lie algebras. More recently, Winter [13] used similar methods in order to study derivations of classical Albert-Zassenhaus algebras. The fact that these methods bear fruit even for infinite dimensional algebras was first observed by Berman [3: 41. This paper is concerned with the determination of derivations of graded Lie algebras with values in graded modules. Our results which are applicable to finitely generated Lie algebras of arbitrary dimension, provide a conceptual framework for the above mentioned papers. We shall be 33