Symmetric matrices, orthogonal Lie algebras and Lie-Yamaguti algebras

Abstract

On the set of symmetric matrices over the field , 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 as derivation algebra. This gives an embedding for . We obtain a sequence of reductive pairs 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 , we use computer algebra to determine the polynomial identities of degree ; we also study the identities relating the bilinear Lie-Yamaguti product with the trilinear product obtained from the Jordan triple product.