On Lusztig's canonical bases of simple Lie algebras

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.