Structure constants for simple Lie algebras from principal $\mathfrak{sl}_2$-triple

TL;DR

This paper develops a method to compute structure constants of simple Lie algebras using principal $rak{sl}_2$-triples, transvectants, and graphical calculus, generalizing known rules from $rak{sl}_2$ to higher types.

Contribution

It introduces a new computational approach for Lie algebra brackets based on invariant theory and graphical calculus, extending classical results from $rak{sl}_2$ to all simple Lie algebras.

Findings

Explicit formulas for $rak{sl}_n$ Lie brackets using transvectants.

Graphical calculus simplifies invariant computations.

Discussion of extensions to other Lie algebra types.

Abstract

For a simple complex Lie algebra $\mathfrak{g}$, fixing a principal $\mathfrak{sl}_2$-triple and highest weight vectors induces a basis of $\mathfrak{g}$ as vector space. For $\mathfrak{sl}_n$, we describe how to compute the Lie bracket in this basis using transvectants. This generalizes a well-known rule for $\mathfrak{sl}_2$ using Poisson brackets and degree 2 monomials in two variables. Our proof method uses a graphical calculus for classical invariant theory. Other Lie algebra types are discussed.