リー三項代数の可解性に関する注意

The class of Lie triple algebras contains Lie algebras and Lie triple systems as special classes. Therefore, the study of Lie triple algebras may need to generalize various concepts in the theory of Lie algebras and Lie triple systems to those in Lie triple algebras. In this paper, we try to introduce the concept of solvability of Lie triple algebras (§1), although their detailed properties are not yet found satisfactorily. We show that both of the standard enveloping Lie algebra and inner derivation algebra of a Lie triplle algebra g are solvable if g is solvable. In §2, the existence of the radical of Lie triple algebras is shown and some results about semi-simple Lie triple algebras are given, under the solvability introduced in S I . The general theory of Lie triple algebras seems to play an important role in the theory of analytic homogeneous systems since it is proved in [1] that every connected and simply connected analytic homogeneous system is characterized by its tangent Lie triple algebra.