On Lusztig's canonical bases of simple Lie algebras

TL;DR

This paper explores Lusztig's canonical bases for simple Lie algebras, demonstrating their construction via multiplicative 2-cocycles in simply laced cases and addressing structure constants in non-simply laced cases.

Contribution

It shows how Lusztig's canonical bases can be derived using 2-cocycles and extends the analysis to non-simply laced root systems.

Findings

Canonical bases can be obtained via multiplicative 2-cocycles.

Explicit structure constants are described for simply laced cases.

Addresses the description of structure constants in non-simply laced cases.

Abstract

Let $\mathfrak{g}$ be a simple Lie algebra over~$\mathbb{C}$ with root system~$\Phi$. In the simply laced case, Frenkel and Kac found a particularly simple construction of~$\mathfrak{g}$, together with a Chevalley basis and explicitly given structure constants, in terms of a certain multiplicative $2$-cocycle $\varepsilon\colon \mathbb{Z} \Phi\times \mathbb{Z}\Phi \rightarrow\{\pm 1\}$. We show that Lusztig's canonical basis of~$\mathfrak{g}$ can also be obtained in this way, for a suitable choice of~$\varepsilon$. We also address the problem of explicitly describing the structure constants when $\Phi$ is not simply laced.