Polynomial Identities and Codimensions of Two- and Three-Dimensional Metabelian Non-Lie Leibniz Algebras

TL;DR

This paper investigates polynomial identities and codimensions of two- and three-dimensional metabelian non-Lie Leibniz algebras, providing explicit bases and analyzing polynomial images over these algebras.

Contribution

It classifies all such algebras in low dimensions, computes their polynomial identities, and explicitly describes the structure of their free algebras.

Findings

Determined finite bases for T-ideals of all classified algebras.

Proved the image of any multilinear polynomial is a vector space.

Explicitly described bases for free graded algebras.

Abstract

Over an arbitrary field, we conduct a comprehensive study of the polynomial identities and codimensions of two- and three-dimensional metabelian non-Lie Leibniz algebras. In addition, we compute the images of multihomogeneous polynomials on two-dimensional Leibniz algebras and, as a consequence, prove that the image of any multilinear polynomial evaluated on such algebras is always a vector space. Our analysis includes the three nontrivial isomorphism classes in dimension two and the ten isomorphism classes in dimension three, all of which are metabelian. In particular, we determine finite bases for their corresponding $T$-ideals and provide explicit bases for the associated relatively free graded algebras.