A polynomial identity for the bilinear operation in Lie-Yamaguti algebras

TL;DR

This paper uses computer algebra to identify a new degree-8 multilinear polynomial identity for the bilinear operation in Lie-Yamaguti algebras, which is not derived from anticommutativity.

Contribution

It explicitly finds a degree-8 identity for Lie-Yamaguti algebras, expanding understanding of their algebraic structure beyond known identities.

Findings

Identifies a degree-8 polynomial identity for Lie-Yamaguti algebras.

Shows no such identities exist in degrees less than 8.

Provides an explicit alternating sum form of the identity.

Abstract

We use computer algebra to demonstrate the existence of a multilinear polynomial identity of degree 8 satisfied by the bilinear operation in every Lie-Yamaguti algebra. This identity is a consequence of the defining identities for Lie-Yamaguti algebras, but is not a consequence of anticommutativity. We give an explicit form of this identity as an alternating sum over all permutations of the variables in a polynomial with 8 terms. Our computations also show that such identities do not exist in degrees less than 8.