论文标题
两个平方零自偏之间或偏斜的内态性和偏斜的总和
Sums of two square-zero selfadjoint or skew-selfadjoint endomorphisms
论文作者
论文摘要
令$ v $为字段$ \ mathbb {f} $上的有限维矢量空间，配备了对称或交替的非脱位双线性表格$ b $。当$ \ mathbb {f} $的特征不是$ 2 $时，我们表征了$ u $ $ v $的$ u $ u $ u $ u = u = a_1+a_1+a_2 $，对于某些一对$ $（a_1，a_1，a_2）$ b $ -b $ -selpadjoint（分别分别$（a_1）^2 =（a_2）^2 = 0 $。在特征$ 2 $的情况下，我们获得了$ v $的内态性的类似分类，当$ b $交替时，将$ v $的两个平方零$ b $ b $ b $ b $ b $ b $ b $ b $ b $ v $分类为$ v $（每当$ b $ a nevernate）时，每当$ b（x，x，x，x，x，x）= 0 $ x $ x $ x $ x $ x $ x。 Finally, if the field $\mathbb{F}$ is equipped with a non-identity involution, we characterize the pairs $(h,u)$ in which $h$ is a Hermitian form on a finite-dimensional space over $\mathbb{F}$, and $u$ is the sum of two square-zero $h$-selfadjoint endomorphisms.
Let $V$ be a finite-dimensional vector space over a field $\mathbb{F}$, equipped with a symmetric or alternating non-degenerate bilinear form $b$. When the characteristic of $\mathbb{F}$ is not $2$, we characterize the endomorphisms $u$ of $V$ that split into $u=a_1+a_2$ for some pair $(a_1,a_2)$ of $b$-selfadjoint (respectively, $b$-skew-selfadjoint) endomorphisms of $V$ such that $(a_1)^2=(a_2)^2=0$. In the characteristic $2$ case, we obtain a similar classification for the endomorphisms of $V$ that split into the sum of two square-zero $b$-alternating endomorphisms of $V$ when $b$ is alternating (an endomorphism $v$ is called $b$-alternating whenever $b(x,v(x))=0$ for all $x \in V$). Finally, if the field $\mathbb{F}$ is equipped with a non-identity involution, we characterize the pairs $(h,u)$ in which $h$ is a Hermitian form on a finite-dimensional space over $\mathbb{F}$, and $u$ is the sum of two square-zero $h$-selfadjoint endomorphisms.
