Polynomial identities and images of polynomials on null-filiform Leibniz algebras

TL;DR

This paper investigates polynomial identities and images on null-filiform Leibniz algebras, providing explicit bases, conditions for images, and confirming conjectures in this algebraic context.

Contribution

It offers a finite minimal basis for identities and characterizes images of polynomials on null-filiform Leibniz algebras, including infinite-dimensional cases.

Findings

Finite minimal basis for polynomial identities of $L_n$

Images of multihomogeneous polynomials are subspaces under certain conditions

The L'vov-Kaplansky conjecture holds for $L_n$

Abstract

In this paper we study identities and images of polynomials on null-filiform Leibniz algebras. If $L_n$ is an $n$-dimensional null-filiform Leibniz algebra, we exhibit a finite minimal basis for $\mbox{Id}(L_n)$, the polynomial identities of $L_n$, and we explicitly compute the images of multihomogeneous polynomials on $L_n$. We present necessary and sufficient conditions for the image of a multihomogeneous polynomial $f$ to be a subspace of $L_n$. For the particular case of multilinear polynomials, we prove that the image is always a vector space, showing that the analogue of the L'vov-Kaplansky conjecture holds for $L_n$. We also prove similar results for an analog of null-filiform Leibniz algebras in the infinite-dimensional case.