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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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