Upper Triangular Matrices with Superinvolution: Identities and Images of Multilinear Polynomials

In this paper, we investigate the algebra of upper triangular matrices UTₙ(F) U T n ( F ) , endowed with a Z₂ Z 2 -grading (i.e., a 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 shown in the framework of Aljadeff et al. (Proc Am Math Soc 145(5):1843–1857, 2017). We provide a complete description of the identities of UT₄(F) U T 4 ( F ) , where the grading is induced by the sequence $$(0,1,0,1)$$ ( 0 , 1 , 0 , 1 ) and the superinvolution is the super-symplectic one. This work extends previous classifications obtained for the cases $$ n = 2 $$ n = 2 and $$ n = 3 $$ n = 3 , and contributes to the study of an open problem for n ≥ 4 n ≥ 4 . In the final part of the paper, we investigate the image of multilinear polynomials on the superalgebra UTₙ(F) U T n ( F ) with superinvolution, showing that the image is a vector space if and only if n ≤ 3 n ≤ 3 , thereby contributing to an analogue of the L’vov–Kaplansky conjecture in this context.