Images of graded polynomials on matrix algebras (Q2097263)

The paper is related to the famous open L'vov-Kaplansky Conjecture, that asks whether the image of a multilinear polynomial on the full matrix algebra \(M_n(K)\) is a vector subspace. The authors study images of \textit{graded} polynomials on full matrix algebras, where the matrix algebra \(M_n(K)\) over a field \(K\) is endowed with its canonical \(\mathbb Z_n\)-grading (Vasilovsky's grading). They explicitly determine the possibilities for the \textit{linear span} of the image of a multilinear graded polynomial over the field \(\mathbb Q\) of rational numbers. Namely, they prove that the \textit{linear span} of a multilinear graded polynomial on a \(\mathbb Q\)-algebra \(A\) is one of the following: \(A_g\), for some \(g \in \mathbb Z_n\), \(\mathbb Q\) (viewed as the set of scalar matrices), the set of trace zero diagonal matrices, or \(\{0\}\). In light of this result they formulate an analogue of the L'vov-Kaplansky conjecture about images of multilinear graded polynomials on \(n \times n\) matrices, where \(n\) is a prime number. The authors confirm this conjecture for polynomials of degree 2 over \(M_n(K)\) when \(K\) is a quadratically closed field of characteristic zero or greater than \(n\) and for polynomials of arbitrary degree over matrices of order 2. They also determine all the possible images of semi-homogeneous graded polynomials evaluated on \(M_2(K)\). There is a recent close research, studying the images of graded polynomials on upper triangular matrices [\textit{P. Fagundes} and \textit{P. Koshlukov}, ``Images of multilinear graded polynomials on upper triangular matrix algebras'', Canad. J. Math. (to appear)].