Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2312.15226

The Table of the Structure Constants for the Complex Simple Lie Algebra of Type G_2 and Chevalley Commutator Formulas in the Chevalley Group of Type G_2 over a Field

Sergey G. Kolesnikov
View original source ↗Read source document ↗Export metadata ↓

Abstract

This article is the second in the series and is devoted to the type G_2. The work consists of two parts. In the first part we calculate the structure constants of the complex simple Lie algebra of type G_2. All structure constants are represented as functions of the structure constants corresponding to extraspecial pairs. The results obtained are used to calculate the commutator Chevalley formulas for [x_r(u),x_s(y)], when the sum r+s is a root. Further, in the second part there is a table of structure constants and Chevalley commutator formulas in the special case, when all structure constants corresponding to extraspecial pairs are positive.

Explore related subjects

math.GR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sergey G. Kolesnikov. 2023-12-23. The Table of the Structure Constants for the Complex Simple Lie Algebra of Type G_2 and Chevalley Commutator Formulas in the Chevalley Group of Type G_2 over a Field. https://arxiv.org/abs/2312.15226

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Subindices and subfactors of $\mathbb{Z}_n$ and $k$-index stability of finite groups

We study subindices, subfactors, and index stability in the cyclic group $\mathbb{Z}_n$. We prove several theorems that not only confirm a conjecture and resolve some open problems about index stability of such groups, but also provide basic tools for the characterization of finite $k$-index stable groups. As a consequence, we completely characterize all 2-element index stable subsets of $\mathbb{Z}_n$, obtain an exact closed formula for their density, and determine all $n$ for which every 2-subset is index unstable. Finally, we present some problems and a research project extending the study to 3-subsets and general $k$-subsets.

math.GR↗
arXiv2026

Virtually generating graphs of pro-$p$ groups

We study the virtually generating graph of pro-$p$ groups. We show that various classes of pro-$p$ groups have connected virtually generating graph and we bound its diameter in these cases; e.g.\ compact subgroups of analytic groups over local fields and the Nottingham group.

math.GR↗
arXiv2026

Surface subgroups of Baumslag doubles along short words

If $U$ is a minimal, diskbusting, finite list of words in a free group $F_n$ of rank $n$ such that the sum of the lengths of words in $U$ is at most $2n+4$, we prove that the natural presentation complex of the Baumslag double of $F_n$ along $U$ virtually contains a $π_1$-injective embedded closed hyperbolic surface. This verifies the Tiling Conjecture of Kim and Wilton for this type of lists of words, and in particular, implies that the corresponding Baumslag double contains a hyperbolic surface subgroup.

math.GR↗