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

2025/09/11 by Campedel, Elena, Fagundes, Pedro, Ioppolo, Antonio
#16R10 #16R50 (Primary) 16W10 #16W50 (Secondary) #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2509.09428

Abstract

In this paper we consider the algebra of upper triangular matrices UT_n(F), endowed with a ℤ₂-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 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 addresses an open problem for n ≥ 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 ≤ 3, thus contributing to an analogue of the L'vov-Kaplansky conjecture in this context.

Citations

• Values of multilinear graded *-polynomials on upper triangular matrices of small dimension
• Graded identities with involution for the algebra of upper triangular matrices
• Commutators and images of noncommutative polynomials

Related

• Superinvolutions on block-triangular matrix algebras
• Values of multilinear graded *-polynomials on upper triangular matrices of small dimension
• Polynomial identities with involution for the algebra of 3 × 3 upper triangular matrices
• Images of linear polynomials on upper triangular matrix algebras
• The images of multilinear polynomials on strictly upper triangular matrices