Lie Triple Systems, Restricted Lie Triple Systems, and Algebraic Groups

We define a restricted structure for Lie triple systems in the characteristic p > 2 setting, akin to the restricted structure for Lie algebras, and initiate a study of a theory of restricted modules. In general, Lie triple systems have natural embeddings into certain canonical Lie algebras, the so-called “standard” and “universal” embeddings, and any Lie triple system can be shown to arise precisely as the −1-eigenspace of an involution (an automorphism which squares to the identity) on some Lie algebra. We specialize to Lie triple systems which arise as the differentials of involutions on simple, simply connected algebraic groups over algebraically closed fields of characteristic p. Under these hypotheses we completely classify the universal and standard embeddings in terms of the Lie algebra and its universal central extension.