Notes on Chevalley Groups and Root Category I

TL;DR

This work builds Chevalley groups from the root category $\mathcal{R}$ by leveraging Hall polynomials and the Ringel–Peng–Xiao framework to produce a Lie-theoretic foundation $\mathfrak{g}_{\mathbb{K}}$ and a group $G(\mathcal{R})$ generated by unipotents $E_{[M]}(t)$. It establishes the Bruhat decomposition, analyzes parabolic and Levi subgroups, and proves simplicity (with four small-field exceptions), while connecting to Steinberg theory to show $\operatorname{Lie}(G) \cong \mathfrak{g}_{\mathbb{K}}$ for algebraically closed fields. For finite fields, it provides an explicit order formula $|G| = \frac{1}{d} q^{r} (q-1)^{m} \sum_{w\in W} q^{\ell(w)}$ and complements this with AR-quiver–based refinements. Overall, the paper unifies triangulated-category constructions with classical Chevalley theory, yielding a self-contained framework for both the structure and size of Chevalley groups arising from root categories.

Abstract

Based on the construction of simple Lie algebras via root category and following Chevalley's results, we construct Chevalley groups from the root category. Then we prove the Bruhat decomposition and the simplicity of the Chevalley groups, and calculate the orders of finite Chevalley groups.

Theorems & Definitions (112)

• Definition 2.1
• Lemma 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Theorem 2.6
• Proposition 2.7
• Lemma 2.8
• Proposition 3.1
• Definition 3.2