Upper triangular matrices with superinvolution: identities and images of multilinear polynomials

Abstract

In this paper we consider the algebra of upper triangular matrices UT$_n(F)$, endowed with a $\mathbb{Z}_2$-grading (superalgebra) and equipped with a superinvolution. These structures naturally arise in the context of Lie and Jordan superalgebras and play a central role in the theory of polynomial identities with involution, as showed in the framework developed by Aljadeff, Giambruno, and Karasik in [2]. We provide a complete description of the identities of UT$_4(F)$, where the grading is induced by the sequence $(0,1,0,1)$ and the superinvolution is the super-symplectic one. This work extends previous classifications obtained for the cases $n = 2$ and $n = 3$, and addresses an open problem for $n \geq 4$. In the last part of the paper, we investigate the image of multilinear polynomials on the superalgebra UT$_n(F)$ with superinvolution, showing that the image is a vector space if and only if $n \leq 3$, thus contributing to an analogue of the L'vov-Kaplansky conjecture in this context.