Symmetric matrices, orthogonal Lie algebras and Lie-Yamaguti algebras

Abstract

On the set Hn(K) of symmetric n × n matrices over the field K 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 so(n, K) as derivation algebra. This gives an embedding so(n, K) ⊂ so(N, K) for N = n+1 2 − 1. We obtain a sequence of reductive pairs (so(N, K), so(n, K)) 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 n ≤ 4, we use computer algebra to determine the polynomial identities of degree ≤ 6; we also study the identities relating the bilinear Lie-Yamaguti product with the trilinear product obtained from the Jordan triple product.

The classical Lie-Yamaguti Yang-Baxter equation and Lie-Yamaguti bialgebras

“…) A pre-Lie-Yamaguti algebra is a vector space A with a bilinear operation * : (7) and {x, y, z} D := {z, y, x} − {z, x, y} + (y, x, z) − (x, y, z), (8) respectively. Here (•, •, •) denotes the associator: (x, y, z) := (x * y) * z − x * (y * z).…”

“…This kind of algebraic structures has attracted much attention recently. For instance, Benito and his colleagues investigated Lie-Yamaguti algebras related to simple Lie algebras of type G 2 [8] and afterwards, they explored orthogonal and irreducible Lie-Yamaguti algebras in [7] and [9,10] respectively. Sheng and the first author focused on linear deformations, product structures and complex structures on Lie-Yamaguti algebras in [21] and later, relative Rota-Baxter operators and pre-Lie-Yamaguti algebras were introduced in [22].…”

Cohomology and deformations of Relative Rota-Baxter operators on Lie-Yamaguti algebras

“…Since then, this system is called a Lie-Yamaguti algebra, which has attracted much attention and is widely investigated recently. For instance, Benito and his collaborators deeply explored irreducible Lie-Yamaguti algebras and their relations with orthogonal Lie algebras [3,4,5,6]. Deformations and extensions of Lie-Yamaguti algebras were examined in [19,20,34,35].…”

Cohomology and relative Rota-Baxter-Nijenhuis structures on LieYRep pairs

“…Afterwards, Benito, Elduque, and Mart ín-Herce explored irreducible Lie-Yamaguti algebras in [8,9]. More recently, Benito, Bremmer, and Madariaga examined orthogonal Lie-Yamaguti algebras in [6]. Yamaguti introduced representations and established cohomology theory of Lie-Yamaguti algebras in [26,27] during from 1950's to 1960's.…”