Values of multilinear graded *-polynomials on upper triangular matrices of small dimension

2023/09/23 by Fagundes, Pedro · 1 citation
#FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2309.13437

Abstract

Let F be an algebraically closed field of characteristic different from 2. We show that the images of multilinear *-polynomials on UT₂ are homogeneous vector spaces. An analogous result holds for UT₃ endowed with non-trivial grading. We further show that these results are optimal, in the following sense: there exist multilinear +graded polynomials whose image on UT_n (n≥ 3) with the trivial grading is not a vector space, and whose image on (UT_n) (n≥ 4) with the ℤ_n-grading is also not a vector space. In particular, an analog of the L'vov-Kaplansky conjecture can not be expected in the setting of algebras with (graded) involutions.

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