Counting gradings on Lie algebras of block-triangular matrices

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications Pub Date : 2024-10-05 DOI:10.1016/j.laa.2024.10.002

Diogo Diniz , Alex Ramos Borges , Eduardo Fonsêca

引用次数: 0

Abstract

We study the number of isomorphism classes of gradings on Lie algebras of block-triangular matrices. Let G be a finite abelian group, for

m \in N^{s}

we determine the number

E^{(-)} (G, m)

of isomorphism classes of elementary G-gradings on the Lie algebra

U T {(m)}^{(-)}

of block-triangular matrices over an algebraically closed field of characteristic zero. We study the asymptotic growth of

E^{(-)} (G, m)

and as a consequence prove that the

E^{(-)} (G, \cdot)

determines G up to isomorphism. We also study the asymptotic growth of the number

N^{(-)} (G, m)

of isomorphism classes of G-gradings on

U T {(m)}^{(-)}

and prove that

N^{(-)} (G, m)) \sim | G | E^{(-)} (G, m)

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块三角矩阵的李代数分级计数

我们研究了块三角矩阵的李代数上等级的同构类数。让 G 是一个有限无性群，对于 m∈Ns ，我们确定了在特征为零的代数闭域上的块三角形矩阵的李代数 UT(m)(-) 上的基本 G 级数的同构类数 E(-)(G,m)。我们研究了 E(-)(G,m)的渐近增长，并由此证明 E(-)(G⋅)决定 G 直到同构。我们还研究了UT(m)(-)上 G-gradings 的同构类数 N(-)(G,m)的渐近增长，并证明了 N(-)(G,m))∼|G|E(-)(G,m)。

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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来源期刊

Linear Algebra and its Applications 数学-应用数学

CiteScore

2.20

自引率

9.10%

发文量

333

审稿时长

13.8 months

期刊介绍： Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.