Filtered algebras and representations of Lie algebras
Filtered algebras and representations of Lie algebras
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DOI：
10.1090/s0002-9947-1961-0130900-1
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发表时间：
1961-03
影响因子：
1.3
通讯作者：
Author R. Sridharan;R. Sridharan
中科院分区：
文献类型：
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作者：
Author R. Sridharan;R. Sridharan
导论.有一个普遍的问题，有多少可以说是关于一个过滤对象通过其相关的分级对象的知识。我们在这里考虑这个一般问题的一个特殊情况。我们取自由K-模L的对称代数S（L），寻找其相关分次代数同构于S（L）的滤子K-代数。一些这样的代数是已知的。事实上，如果g表示L上的任意李代数，“庞加莱-维特定理”断言g的泛包络代数就是这样一个李代数。事实证明，这“几乎”给出了我们问题的一般解。事实上，我们所寻求的代数是通常包络代数的适当推广，并且事实上可以被定义为L上李代数的某些“广义表示”的泛对象。其余的结果是关于这些代数的上同调的。对于交换环K上的李代数g和它的标准复形上的2-上圈f，我们定义在？第一个是frepresentation。g的通常表示对应于f= 0的情况。我们介绍？2.过滤K-代数gf是f-表示的一个通用模型。我们推出的“庞加莱-维特定理”（定理2.6）为gf作为一个简单的后果，通常的庞加莱-维特定理，证明在[1，第271页]。如果g是K-free的，则存在分次K-代数同构4 t 'f：S（g）-*E0（gf），其中S（g）表示K-模g的对称代数，E0（gf）表示与gf相关联的分次代数（定理2.5）。在哪？3.对一个固定的分次K-代数S，我们定义了对象为对（A，41 A）的范畴，其中A是一个滤子K-代数，IPA：SEO（A）是分次代数的同构，并且其映射是以一种明显的方式定义的。若S是自由K-模L的对称代数，则这类对象与对（g，f）的同构类之间存在1-1对应，其中g是L上的李代数，fH 2（g，K）.对于类f中的一个上圈f，对（gf，ifr）是对应于（g，f）的类中的一个对象（定理3.1）。第四节研究有限维李代数g的某些通常的同调群和上同调群的计算。这相当于对f= 0的g&的研究。这些计算将在下一个
Introduction. There is a general question as to how much can be said about a filtered object through the knowledge of its associated graded object. We consider here a particular case of this general problem. We take the symmetric algebra S(L) of a free K-module L and look for filtered K-algebras whose associated graded algebras are isomorphic to S(L). Some such algebras are already known. In fact if g denotes an arbitrary Lie algebra on L, the "Poincar6-Witt Theorem" asserts that the universal enveloping algebra of g is one such. It turns out that this gives "almost" a general solution of our problem. Indeed, the algebras we seek are suitable generalizations of the usual enveloping algebras and can in fact be defined as universal objects for certain "generalized representations" of Lie algebras on L. The rest of our results are on the cohomology of these algebras. For a Lie algebra g over a commutative ring K and a 2-cocycle f on its standard complex with values in K, we define in ?1 the notion of an frepresentation. The usual representations of g correspond to the case f= 0. We introduce in ?2 the filtered K-algebra gf which is a universal model for f-representations. We deduce the "Poincare-Witt Theorem" (Theorem 2.6) for gf as an easy consequence of the usual Poincare-Witt Theorem, proved in [1, p. 271]. It is then clear that if g is K-free, there is a graded K-algebra isomorphism 4t'f: S(g) -*E0(gf), where S(g) denotes the symmetric algebra of the K-module g and EO(gf) the graded algebra associated with gf (Theorem 2.5). In ?3, we define, for a fixed graded K-algebra S, the category whose objects are pairs (A, 41A), where A is a filtered K-algebra, IPA: SEO(A) an isomorphism of graded algebras and whose maps are defined in an obvious manner. If S is the symmetric algebra of a free K-module L, then there is a 1-1 correspondence between isomorphism classes of such objects and pairs (g, f), where g is a Lie algebra on L and f H2(g, K). For a cocyclef in the class f, the pair (gf, ifr) is an object in the class corresponding to (g, f) (Theorem 3.1). The fourth section is devoted to the computations of certain of the usual homology and cohomology groups of a finite dimensional Lie algebra g. This amounts to a study of g& for f= 0. These computations are used in the next