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

Abstract 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.