A construction of Lie algebras from a class of ternary algebras
A construction of Lie algebras from a class of ternary algebras
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从一类三元代数构造李代数
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发表时间：
1971
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通讯作者：
J. Faulkner
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作者：
J. Faulkner
定义了一类具有三元复合和交错双线性型的代数。从这类的一个成员的李代数的结构，并证明了李代数是简单的，如果形式是非退化的。在基环含有1/2的情况下，得到了这样构造的李代数作为三维单李代数的模的结构的一个特征。最后，一些李代数被确定;特别是，E8型李代数被获得。由J. Tits [7]和M. Koecher [4]在这两种代数的研究中都很有用。本文给出了由具有满足某些公理的斜双线性型的三元代数构造李代数的类似结果。这些三元代数是文[1]中所考虑的Freudenthal三元系的一个变体。我们得到的大多数结果与TitsKoecher构造的结果相似（见[3，第VIII章]）。在哪？1中定义了三元代数，得到了关于三元代数的一些基本结果，并给出了这类代数的两个例子。在哪？2.构造了单李代数当且仅当斜双线性型是非退化的。在哪？3.在基环含有1/2的情况下，我们给出了由我们构造的李代数作为三维单李代数模的结构的一个特征。最后，在？4.我们从？1.特别是，我们表明，我们可以从一个56维空间，这是一个模的李代数E7型的E8型李代数。H. Freudenthal在[2]。1.一类三元代数。我们感兴趣的是任意可换结合环D上的模TM，它具有交错双线性形式和三元积，满足：（T1）= + z，x，y，ze M;（T2）= + x，x，y，z ∈ 9;（T3），w> =，z> +，x，y，z，w ∈ 9;编辑于1970年3月12日收到，修订版于1970年6月19日收到。AMS 1969主题分类。第1730章.
A class of algebras with a ternary composition and alternating bilinear form is defined. The construction of a Lie algebra from a member of this class is given, and the Lie algebra is shown to be simple if the form is nondegenerate. A characterization of the Lie algebras so constructed in terms of their structure as modules for the three-dimensional simple Lie algebra is obtained in the case the base ring contains 1/2. Finally, some of the Lie algebras are identified; in particular, Lie algebras of type E8 are obtained. A construction of Lie algebras from Jordan algebras discovered independently by J. Tits [7] and M. Koecher [4] has been useful in the study of both kinds of algebras. In this paper, we give a similar construction of Lie algebras from a ternary algebra with a skew bilinear form satisfying certain axioms. These ternary algebras are a variation on the Freudenthal triple systems considered in [1]. Most of the results we obtain for our construction are parallel to those for the TitsKoecher construction (see [3, Chapter VIII]). In ?1, we define the ternary algebras, derive some basic results about them, and give two examples of such algebras. In ?2, the Lie algebras are constructed and shown to be simple if and only if the skew bilinear form is nondegenerate. In ?3, we give a characterization, in the case the base ring contains 1/2, of the Lie algebras obtained by our construction in terms of their structure as modules for the threedimensional simple Lie algebra. Finally, in ?4, we identify some of the simple Lie algebras obtained by our construction from the examples of ?1. In particular, we show that we can construct a Lie algebra of type E8 from a 56-dimensional space which is a module for a Lie algebra of type E7. A similar construction was given by H. Freudenthal in [2]. 1. A class of ternary algebras. We shall be interested in a module TM over an arbitrary commutative associative ring D with 1 which possesses an alternating bilinear form and a ternary product which satisfy (TI) = + z for x,y, ze M; (T2) = + x for x, y, z e 9; (T3) , w> = , z> + for x, y, z, w e 9; Received by the editors March 12, 1970 and, in revised form, June 19, 1970. AMS 1969 subject classifications. Primary 1730.