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

In this paper, we investigate the algebra of upper triangular matrices UTn(F), endowed with a Z2-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), 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 contributes to the study of an open problem for n≥4. In the final part of the paper, we investigate the image of multilinear polynomials on the superalgebra UTn(F) with superinvolution, showing that the image is a vector space if and only if n≤3, thereby contributing to an analogue of the L’vov–Kaplansky conjecture in this context.