发表机构
Stockholm University; Taras Shevchenko National University of Kyiv（斯德哥尔摩大学; 基辅塔拉斯舍甫琴科国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解决了三个例外简单有限维李超代数的极大分次子代数分类问题，列出了适用分次并给出所有极大分次子代数，且通过判据提升为极大，并补充了分裂分类的第六种情形。
AI 中文摘要
Dynkin 对简单有限维李代数的极大子代数进行了分类。其超版本问题对于矩阵李超代数已由 Shchepochkina 解决，而对于具有多项式系数并带有 Weisfeiler 分次的简单向量李超代数，在 Leites 和 Shchepochkina 的姊妹篇论文中已解决，但遗留了三个未解决的问题：三个简单有限维李超代数 $\mathfrak{osp}_a(4|2)$、$\mathfrak{ag}(2)$ 和 $\mathfrak{ab}(3)$，它们以多种方式向量化，且未被一般论证所覆盖。本文针对**分次**子代数解决了这三种情形。对于这三个李超代数中的每一个，我们列出了该论文方法适用的所有 $\mathbb{Z}$-分次——$\mathfrak{osp}_a(4|2)$ 有两个，$\mathfrak{ag}(2)$ 有五个，$\mathfrak{ab}(3)$ 有八个——并描述了每一个分次下的所有极大分次子代数，其中半单子代数被显式给出。一个简单的判据将大多数答案从“极大分次”提升为“极大”。每个答案都在 $\mathbb{Q}$ 上计算。两个副产品：向量李超代数的分次子代数分裂为**五**类的经典列表对于深度 $>1$ 的环境是不完整的——必须添加第六类，且它恰好在此处非空；并且位于 $\mathfrak{ag}(2)$ 和 $\mathfrak{ab}(3)$ 内部的 $\mathfrak{osp}_a(4|2)$ 副本的参数 $a$ 被内在地计算出来。
英文摘要
Dynkin classified the maximal subalgebras of the simple finite-dimensional Lie algebras. The super version of his problem is solved, for the matrix Lie superalgebras, by Shchepochkina, and, for the simple vectorial ones with polynomial coefficients taken with their Weisfeiler gradings, in the companion paper by Leites and Shchepochkina --- with three exceptions left open: the three simple finite-dimensional Lie superalgebras $\mathfrak{osp}_a(4|2)$, $\mathfrak{ag}(2)$ and $\mathfrak{ab}(3)$, which are vectorial in more than one way and are not covered by the general argument. Here we settle these three cases for \textbf{graded} subalgebras. For each of the three Lie superalgebras, we list all the $\mathbb{Z}$-gradings to which the method of that paper applies --- two for $\mathfrak{osp}_a(4|2)$, five for $\mathfrak{ag}(2)$, eight for $\mathfrak{ab}(3)$ --- and describe, for every one of them, all the maximal graded subalgebras, the semi-simple ones explicitly. A simple criterion upgrades most of the answers from ``maximal graded\rq\rq\ to ``maximal\rq\rq. Every answer is computed over $\mathbb{Q}$. Two by-products: the classical list of \textbf{five} cases into which the graded subalgebras of a vectorial Lie superalgebra split is incomplete for ambients of depth $>1$ --- a sixth case has to be added, and it is non-empty precisely here; and the parameter $a$ of the copies of $\mathfrak{osp}_a(4|2)$ sitting inside $\mathfrak{ag}(2)$, resp. $\mathfrak{ab}(3)$, is computed intrinsically.
Comments21 pages
