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有限根系和简单李代数的正则分解

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra Pub Date : 2024-11-19 DOI:10.1016/j.jalgebra.2024.10.037

Stepan Maximov

引用次数: 0

摘要

设 g 是特征为 0 的代数闭域上的有限维简单李代数。本文将对 g 及其不可还原根系 Δ 的所有正则分解进行分类。正则分解是指分解 g=g1⊕...⊕gm，其中每个 gi 和 gi⊕gj 都是正则子代数。这样的分解会引起相应根系统的分区，即 Δ=Δ1⊔...⊔Δm，使得所有 Δi 和 Δi⊔Δj 都是封闭的。在本文中，我们证明了 m⩾3 的情况只可能出现在 An 类型的系统中，并用 (n+1) 的 m 分区描述了所有这样的分区。然后，我们将这些结果推广到 g 的正则分解的分类中。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Regular decompositions of finite root systems and simple Lie algebras

Let

g

be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic 0. In this paper we classify all regular decompositions of

g

and its irreducible root system Δ.

A regular decomposition is a decomposition

g = g_{1} \oplus \dots \oplus g_{m}

, where each

g_{i}

and

g_{i} \oplus g_{j}

are regular subalgebras. Such a decomposition induces a partition of the corresponding root system, i.e.

Δ = Δ_{1} ⊔ \dots ⊔ Δ_{m}

, such that all

Δ_{i}

and

Δ_{i} ⊔ Δ_{j}

are closed.

Partitions of Δ with

m = 2

were known before. In this paper we prove that the case

m ⩾ 3

is possible only for systems of type

A_{n}

and describe all such partitions in terms of m-partitions of

(n + 1)

. These results are then extended to a classification of regular decompositions of

g

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来源期刊

Journal of Algebra 数学-数学

CiteScore

1.50

自引率

22.20%

发文量

414

审稿时长

2-4 weeks

期刊介绍： The Journal of Algebra is a leading international journal and publishes papers that demonstrate high quality research results in algebra and related computational aspects. Only the very best and most interesting papers are to be considered for publication in the journal. With this in mind, it is important that the contribution offer a substantial result that will have a lasting effect upon the field. The journal also seeks work that presents innovative techniques that offer promising results for future research.