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

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 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. (c) 2023 Elsevier Inc. All rights reserved.