Symmetric matrices, orthogonal Lie algebras and Lie-Yamaguti algebras

Resumen

On the set (Formula presented.) of symmetric (Formula presented.) matrices over the field (Formula presented.) , we can define various binary and ternary products which endow it with the structure of a Jordan algebra or a Lie or Jordan triple system. All these nonassociative structures have the orthogonal Lie algebra (Formula presented.) as derivation algebra. This gives an embedding (Formula presented.) for (Formula presented.). We obtain a sequence of reductive pairs (Formula presented.) that provides a family of irreducible Lie-Yamaguti algebras. In this paper, we explain in detail the construction of these Lie-Yamaguti algebras. In the cases (Formula presented.) , we use computer algebra to determine the polynomial identities of degree (Formula presented.) ; we also study the identities relating the bilinear Lie-Yamaguti product with the trilinear product obtained from the Jordan triple product.