Upper triangular matrices with superinvolution: identities and images of multilinear polynomials
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arXiv
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| Natura: | Preprint |
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2025
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| author | Campedel, Elena Fagundes, Pedro Ioppolo, Antonio |
| author_facet | Campedel, Elena Fagundes, Pedro Ioppolo, Antonio |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09428 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Upper triangular matrices with superinvolution: identities and images of multilinear polynomials Campedel, Elena Fagundes, Pedro Ioppolo, Antonio Rings and Algebras 16R10, 16R50 (Primary) 16W10, 16W50 (Secondary) 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. |
| title | Upper triangular matrices with superinvolution: identities and images of multilinear polynomials |
| topic | Rings and Algebras 16R10, 16R50 (Primary) 16W10, 16W50 (Secondary) |
| url | https://arxiv.org/abs/2509.09428 |