论文标题
在两个发电机上可解的谎言代数的nilradical的尼尔氏脉络度
Nilpotency degree of the nilradical of a solvable Lie algebra on two generators
论文作者
论文摘要
给定序列$ \ vec d =（d_1，\ dots，d_k）$的自然数，我们考虑谎言subergebra $ \ mathfrak {h} $ of $ \ mathfrak {gl} {gl} {d，\ \ m mathbb {f}）$由两个块上三角矩阵$ d $和$ e $根据$ \ vec d $生成，并研究计算nilradical $ \ mathfrak {n} $ of $ \ mathfrak {h Mathfrak {h h} $的nilradical $ \ mathfrak {n} $的nilpotency $ m $。当$ d $和$ e $属于某种矩阵时，我们会在尝试对某些可解决的Lie代数的不可分解的模块进行分类时自然而然地获得一个完整的答案。 我们对$ m $的确定取决于$ e $的对称性相对于$ \ mathfrak {sl}（d）$的对称性。 $ m $仅取决于这种对称性的证据是漫长而精致的。 作为我们对$ \ mathfrak {h} $和$ \ mathfrak {n} $的调查的直接应用，我们对所有单个单性模块进行了完整分类，以扩展免费的$ \ ell $ \ ell $ step nilpotent lie eLgebra $ n $ n $ n $ n $ Generals on $ \ the Mathbb {f} f} $ {f} $关闭。
Given a sequence $\vec d=(d_1,\dots,d_k)$ of natural numbers, we consider the Lie subalgebra $\mathfrak{h}$ of $\mathfrak{gl}(d,\mathbb{F})$, where $d=d_1+\cdots +d_k$ and $\mathbb{F}$ is a field of characteristic 0, generated by two block upper triangular matrices $D$ and $E$ partitioned according to $\vec d$, and study the problem of computing the nilpotency degree $m$ of the nilradical $\mathfrak{n}$ of $\mathfrak{h}$. We obtain a complete answer when $D$ and $E$ belong to a certain family of matrices that arises naturally when attempting to classify the indecomposable modules of certain solvable Lie algebras. Our determination of $m$ depends in an essential manner on the symmetry of $E$ with respect to an outer automorphism of $\mathfrak{sl}(d)$. The proof that $m$ depends solely on this symmetry is long and delicate. As a direct application of our investigations on $\mathfrak{h}$ and $\mathfrak{n}$ we give a full classification of all uniserial modules of an extension of the free $\ell$-step nilpotent Lie algebra on $n$ generators when $\mathbb{F}$ is algebraically closed.
