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Lie triple systems, restricted Lie triple systems, and algebraic groups. (English) Zbl 0987.17011
A Lie triple system may be defined as the odd part of a \({\mathbb Z}_2\)-graded Lie algebra, with the ternary operation given by \([xyz]=[[x,y],z]\) (inside the Lie algebra). The aim of the paper is to study the Lie triple systems related to the pairs \((G,\theta)\), where \(G\) is a simple and simply connected algebraic group over an algebraically closed field of prime characteristic \(p>2\), endowed with an involution \(\theta\).
Among the \({\mathbb Z}_2\)-graded Lie algebras whose odd part is a specific Lie triple system, there is a sort of minimal one: the standard enveloping Lie algebra, and a maximal one: the universal enveloping Lie algebra. These are completely determined for the Lie triple systems associated to the pairs \((G,\theta)\) above, in terms of the Lie algebra of \(G\) and its universal central extension. The main problems arise for cases \(A_2\) and \(G_2\) in characteristic \(3\).
Also the notion of restricted Lie triple systems is developed, as well as those of modules and restricted modules for a (restricted) Lie triple system, relating these concepts with the corresponding ones for the enveloping Lie algebras.
Several interesting open questions about the development of a restricted cohomology theory, and questions regarding the relationship between the representation theory of the pairs \((G,\theta)\) and that of the associated Lie triple systems are posed.

MSC:
17B45 Lie algebras of linear algebraic groups
17A40 Ternary compositions
17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
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