Symmetric matrices, orthogonal Lie algebras and Lie-Yamaguti algebras

Abstract

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.