L’vov–Kaplansky conjecture on images of multilinear polynomials over matrix algebras

Establish whether, for every field K and every integer n≥1, the image f(M_n(K)) of any multilinear noncommutative polynomial f(x_1,…,x_m) evaluated on the full matrix algebra M_n(K) is a vector subspace of M_n(K), thereby resolving the L’vov–Kaplansky conjecture.

Background

The paper discusses images of polynomials as a generalization of polynomial identities and highlights the L’vov–Kaplansky conjecture, which predicts that the image of any multilinear polynomial on M_n(K) is a vector subspace. The authors note that the conjecture has only been resolved in special cases (n=2 and m=2) and remains open in general.

Within the present work, the authors establish an analogue for two-dimensional Leibniz algebras by proving that the image of any multilinear polynomial evaluated on such algebras is always a vector space, providing supportive evidence and motivation for the broader open conjecture.