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上三角形矩阵上的齐次对合

Homogeneous involutions on upper triangular matrices

Thiago Castilho de Mello

发表信息：
Arch. Math. 118, 365-374 (2022)
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摘要：

令$K$是特征不同于2的域，令$G$是群。如果$K$上的$n\times n$上三角形矩阵的代数$UT_n$被赋予一个$G$ -分次$Γ: UT_n=\oplus_{g\in G}A_g$，我们给出了$Γ$上保证$A$上存在齐次反自同构的充分必要条件，即，一个反自同构$φ$满足$φ(A_g)=A_{θ(g)}$，对于分级的支持度的某个置换$θ$。证明了$UT_n$允许齐次反自同构当且仅当$UT_n$的反射对合是齐次的。证明了如果$UT_n$的一个齐次反自同构由映射$θ$定义，则任何其它的齐次反自同构都由同一映射$θ$定义.

英文摘要：

Let $K$ be a field of characteristic different from 2 and let $G$ be a group. If the algebra $UT_n$ of $n\times n$ upper triangular matrices over $K$ is endowed with a $G$-grading $Γ: UT_n=\oplus_{g\in G}A_g$ we give necessary and sufficient conditions on $Γ$ that guarantees the existence of a homogeneous antiautomorphism on $A$, i.e., an antiautomorphism $φ$ satisfying $φ(A_g)=A_{θ(g)}$ for some permutation $θ$ of the support of the grading. It turns out that $UT_n$ admits a homogeneous antiautomorphism if and only if the reflection involution of $UT_n$ is homogeneous. Moreover, we prove that if one homogeneous antiautomorphism of $UT_n$ is defined by the map $θ$ then any other homogeneous antiautomorphism is defined by the same map $θ$.

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