A construction of Lie algebras by triple systems
A construction of Lie algebras by triple systems
复制标题
三重系构造李代数
DOI：
10.1090/s0002-9947-1975-0393153-5
复制
发表时间：
1975
影响因子：
1.3
通讯作者：
W. Hein
中科院分区：
文献类型：
--
作者：
W. Hein
导论.由弗赖登塔尔、福克纳或克彻构造得到的所有李代数的一个共同特征（参见：[4]、[3]、[7]分别）是在任何这样的代数中有一个X，使得ad X的特征值包含在集合{0，± 1，± &}中（参见{3]，[4]，[7]）。此外，+1和- &特征空间的直接和原来是一个三元系（一个向量空间加上三线性内组成），使？分解为向量空间直和5c（2）© 5 © 5，其中5c（2）是2上所有线性变换的李代数的子代数，%是%的另一个副本。如果G是由Koecher构造得到的，则± H特征空间是{0};如果E是由Faulkner方法得到的，则±1特征空间是一维的。查询特征为01的代数闭域上的单李代数的根的表发现，每个这样的李代数包含元素X，使得ad X的特征空间不满足上述维数限制。这就是本文件的出发点。在§ 1中，我们定义了向量空间5l（ε））S <$S上的一个反交换代数结构，其中S是某个三元系，ε是S的另一个副本，ε是S的一个副本，ε是S的一个副本。(2)线性变换的李代数。假设满足Jacobi恒等式，我们推导出一组恒等式，它们通过Jordan子三元系21（定义见[7]或[8]）和某个三元系8来描述S，Jordan三元系2n在8的基础向量空间上的特殊表示a和从8 × 8到21的双线性映射（如果a是忠实的，则不需要f）。我们把自己限制在21是
Introduction. A common feature of all the Lie algebras obtained by either the Freudenthal, Faulkner or Koecher construction (cf. [4], [3], [7] resp.)is that there is an X in any such algebra Ç such that the eigenvalues of ad X are contained in the set {0, ± 1, ± &} (cf. {3], [4], [7] ). Moreover, the direct sum of the +1 and — & eigenspace turns out to be a triple system (a vector space together with a trilinear inner composition) such that ? decomposes as the vector space direct sum 5ß (2)© 5 © 5 where 5c(2) is a subalgebra of the Lie algebra of all linear transformations on 2 and % is another copy of %. If G is obtained by the Koecher construction, the ± H eigenspaces are {0}; if £ is obtained by the method of Faulkner, the ±1 eigenspaces are one dimensional. Consulting a table of roots of the simple Lie algebras over an algebraically closed field of characteristic zero one finds that each such Lie algebra contains an element X such that the eigenspaces of ad X do not satisfy the above dimension restrictions. This is the point where the present paper starts. In § 1, we define an anticommutative algebra structure on the vector space 5l(£) © S © S where S is a certain triple system, £ another copy of S and ï!(2) a Lie algebra of linear transformations of %. Assuming the Jacobi identity to be satisfied we deduce a set of identities which describes S by a Jordan sub triple system 21 (see [7] or [8] for definition) and a certain triple system 8, a special representation a of the Jordan triple system 2Ï on the underlying vector space of 8 and a bilinear mapping from 8x8 to 21 (/ is not needed if a is faithful). We restrict ourselves to the case where 21 is