On the structure of symplectic ternary algebras
On the structure of symplectic ternary algebras
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辛三元代数的结构
DOI：
10.1016/1385-7258(72)90062-5
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发表时间：
1972
影响因子：
1.7
通讯作者：
J. Ferrar
中科院分区：
文献类型：
--
作者：
J. Faulkner;J. Ferrar
本文所研究的代数是文[3]中给出的三元代数类的推广，而这类代数又是Freudenthal三元系的一个变形[7]。最新的代数，我们称之为辛代数，其优点在于它们是由恒等式定义的，因此允许直和。本文的主要目的是证明半单辛代数是特征为0的单代数的直和，并给出特征为0的代数闭域上的单代数的分类.在这个过程中，我们得出了一些结果的字段的任意特征$12，3。在$1中，我们将辛代数的可解性和半单性的内在概念与从辛代数构造的李三系中的平行概念联系起来。在5.2中，我们利用这个关系证明了特征为0的半单辛代数具有非退化迹形式。这使我们能够推导出第一结构定理。在8.3中，指出了辛代数、Freudenthal三元系和[3]中的三元代数之间的关系。在Q4;通过对[7]中Freudenthal三元系的分类的简化，对具有非零斜形式的平衡单辛代数进行了分类，得到了特征为零的代数闭域上的单辛代数的完全分类（推论4.2）。在整个过程中，我们假设特性0#2，3。
The algebras studied here are a generalization of the class of ternary algebras given in [3], which are, in turn, a variation of Freudenthal triple systems [7]. The advantage of the latest algebras, which we call symplectic algebras, is that they are defined by identities and hence admit direct sums. The main purpose of this paper is to show that semisimple symplectic algebras are the direct sum of simple algebras in characteristic 0 and to give a classification of the simple algebras over algebraically closed fields of characteristic 0. In the process we derive a number of results for fields of arbitrary characteristic $12, 3. In $1, we relate intrinsic notions of solvability and semisimplicity of a symplectic algebra to the parallel notions in a Lie triple system constructed from the symplectic algebra. In 5 2, we use this relation to show that a semisimple symplectic algebra of characteristic 0 has nondegenerate trace form. This allows us to derive the first structure theorem. In 8 3, the relationships among symplectic algebras; Freudenthal triple systems, and the ternary algebras of [3] are indicated. In Q 4; the balanced, simple symplectic algebras with nonzero skew form are classified via a reduction to the classification of Freudenthal triple systems in [7], yielding (Corollary 4.2) complete classification of simple, symplectic algebras over algebraically closed fields of characteristic zero. Throughout we assume characteristic 0# 2, 3.