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Identity classification in triple systems. (English) Zbl 0596.17002
The author classifies nontrivial triple systems $$\mathcal M$$ defined on a finite-dimensional vector space over an algebraically closed field of characteristic zero such that: (i) every left multiplication $$L(x,y)$$, $$x,y\in\mathcal M$$, is a derivation of $$U$$, where $$L(x,y)z=xyz$$ is the triple product of $$\mathcal M$$, (ii) $$\mathcal M$$ is irreducible under $$\operatorname{Der}(\mathcal M)$$ and (iii) $$\mathcal M$$ satisfies an (unspecified) 3-linear identity.
After studying all possible 3-linear identities in general triple systems (via the left ideal structure in the group algebra of $$S_3)$$ the author shows that $$\mathcal M$$ lies in one of 6 classes each defined by identities. Calling left-skew: $$xyz=-yxz$$, left-symmetric: $$xyz=yxz$$, Jacobi: $$xyz+yzx+zxy=0$$, cyclic: $$xyz=yzx$$, these classes are (a) left-skew, (b) left-symmetric, (c) left-skew and Jacobi (Lie triple systems), (d) left-symmetric and Jacobi, (e) Jacobi, (f) left skew and cyclic. For each of these six classes the author obtains a precise classification using the representation theory of Lie algebras. This can be done since for example an immediate consequence of (i) and (ii) is that $$\operatorname{Der}(\mathcal M)$$ is semisimple and $$\mathcal M$$ is a self-dual module.
As a by-product of his methods he gives a quick proof of the well-known classification of flexible Lie-admissible algebras $$A$$ with $$A^-$$ simple.
Reviewer: Erhard Neher

##### MSC:
 17A40 Ternary compositions 17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.) 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 17D25 Lie-admissible algebras 17A20 Flexible algebras
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