On the Lie Triple System and its Generalization
1958; Hiroshima University - Department of Mathematics; Volume: 21; Issue: 3 Linguagem: Inglês
10.32917/hmj/1555639527
ISSN0018-2079
Autores Tópico(s)Mathematics and Applications
ResumoIn a Lie triple system st 1 > (L.t.s.) over a base field (P,2' let D be a vector space generated from the set of linear mappings X-+ I[aibtx] and D' be a vector subspace generated from such set of linear mappings as [abx] =0 for all x in.st, then the factor space ':n(st)=D/D' has the structure of Lie algebra of linear mappings of st (inner derivations algebra of st).The following theorem was first established by N. Jacobson [3] and improved under weaker assumptions than his in [6].THEOREM.L.t.s.st can be 1-to-1 imbedded into a Lie algebra 2 in such a way that the given composition [abc] in st coincides with the product [[ab]c] defined in 2 and 2='.itffi':n(st).2 is called a standard enveloping Lie algebra of st.E. Cartan proved that Lie algebra is semi-simple if and only if the determinant I gi 1 f-ig..not zero.In §l we shall generalize this result to L.t.s., and prove some other prep8rties.In § 2, we shall define the general Lie triple system which has the geometrical meaning, and prove that the general Lie triple system can be imbedded into a Lie algebra.§ 1.Some properties of Lie triple systems.
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