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No abstract

The class of structurable algebras has been defined by Allison [I] We denote by M the subspace of the tensor product S~q6 $ generated by the set IS t@$z -6~$/: $I,$26S }. We set ~=S~0S/~ and on the direct sum of the spaces ~S we define a commutative operation • and an anticormnutative operation [, ] by the rules:

We show that the category of Lie triple systems is equivalent to the category of Z 2 -graded Lie algebras L = L 0 ⊕ L 1 such that L 1 generates L and the second graded cohomology group of L with coefficients in any trivial module is zero. As a corollary we obtain an analogous result for symmetric spaces and Lie groups.

Abstract. Kantor pairs arise naturally in the study of -graded Lie algebras. In this article, we introduce and study Kantor pairs with short Peirce gradings and relate them to Lie algebras graded by the root system of type BC . is relationship allows us to de ne so called Weyl images of short Peirce graded Kantor pairs. We use Weyl images to construct new examples of Kantor pairs, including a class of in nite dimensional central simple Kantor pairs over a eld of characteristic = or , as well as a family of forms of a split Kantor pair of type mathrmE .