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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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References:
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