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On the representation theory of Lie triple systems. (English) Zbl 1012.17001

New results concerning the representation theory and cohomology theory of Lie triple systems are presented in this article. For a Lie triple system \(T\), let \(L_u (T) = \mathfrak L/\mathfrak I\) where \(\mathfrak L\) is the free Lie algebra based on \(T\), and \(\mathfrak I\) is the ideal in \(\mathfrak L\) generated by the elements \([a,b,c]-[[a,b],c]\) for \(a,b,c \in T\). Assume that the characteristic of the base field is not 2, and let \(\theta : L_u (T) \rightarrow L_u (T)\) be the involution with \(-1\) -eigenspace \(T\). The authors consider the precise relationship of the category of \(T\)-modules to the category of modules of \(L_u (T)\) that have a compatible action of \(\theta\). There is a parallel development of a theory of restricted representations of restricted Lie triple systems and also the beginning of a study of the cohomology of Lie triple systems.

MSC:

17A40 Ternary compositions
17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
17B56 Cohomology of Lie (super)algebras
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