<p>In this paper, we investigate the algebra of upper triangular matrices <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ UT_n(F) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <msub> <mi>T</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, endowed with a <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {Z}_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-grading (i.e., a superalgebra) and equipped with a superinvolution. These structures naturally arise in the context of Lie and Jordan superalgebras and play a central role in the theory of polynomial identities with involution, as shown in the framework of Aljadeff et al. (Proc Am Math Soc 145(5):1843–1857, 2017). We provide a complete description of the identities of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ UT_4(F) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <msub> <mi>T</mi> <mn>4</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where the grading is induced by the sequence <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(0,1,0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and the superinvolution is the super-symplectic one. This work extends previous classifications obtained for the cases <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ n = 2 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ n = 3 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, and contributes to the study of an open problem for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$ n \ge 4 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>. In the final part of the paper, we investigate the image of multilinear polynomials on the superalgebra <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$ UT_n(F) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <msub> <mi>T</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with superinvolution, showing that the image is a vector space if and only if <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$ n \le 3 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≤</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, thereby contributing to an analogue of the L’vov–Kaplansky conjecture in this context.</p>

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

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2026年5月13日

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Elena Campedel,
Pedro Fagundes,
Antonio Ioppolo

摘要

In this paper, we investigate the algebra of upper triangular matrices $ UT_n(F) $ $U T_{n} (F)$ , endowed with a $\mathbb {Z}_2$ $Z_{2}$ -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 $ UT_4(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 \ge 4 $ $n \geq 4$ . In the final part of the paper, we investigate the image of multilinear polynomials on the superalgebra $ UT_n(F) $ $U T_{n} (F)$ with superinvolution, showing that the image is a vector space if and only if $ n \le 3 $ $n \leq 3$ , thereby contributing to an analogue of the L’vov–Kaplansky conjecture in this context.