On the construction of semisimple Lie algebras and Chevalley groups

TL;DR

This paper introduces a new, elementary method for constructing semisimple Lie algebras from their root systems, also providing explicit Chevalley bases, building on Lusztig's recent simplifications of Chevalley group constructions.

Contribution

It presents a novel approach to constructing semisimple Lie algebras directly from root systems, yielding explicit Chevalley bases, and simplifies the understanding of their structure.

Findings

New elementary construction of semisimple Lie algebras from root systems

Explicit Chevalley bases derived from the construction

Simplifies the process compared to traditional methods

Abstract

Let $\mathfrak{g}$ be a semisimple complex Lie algebra. Recently, Lusztig simplified the traditional construction of the corresponding Chevalley groups (of adjoint type) using the "canonical basis" of the adjoint representation of~$\mathfrak{g}$. Here, we present a variation of this idea which leads to a new, and quite elementary construction of~$\mathfrak{g}$ itself from its root system. An additional feature of this set-up is that it also gives rise to explicit Chevalley bases of $\mathfrak{g}$.