General Formulas of the Structure Constants in the $\mathfrak{su}(N)$ Lie Algebra

TL;DR

This paper derives simple, closed-form analytic expressions for the symmetric and anti-symmetric structure constants of the ${su}(N)$ Lie algebra, facilitating easier computations in physics.

Contribution

It provides the first explicit formulas for the structure constants of ${su}(N)$, linking generator indices to their non-zero elements without needing generator matrices.

Findings

Explicit formulas involve generator indexes and are easy to evaluate.

Formulas enable analytical and computational applications in physics.

Simplifies calculations of structure constants in ${su}(N)$ algebra.

Abstract

We provide the analytic expressions of the totally symmetric and anti-symmetric structure constants in the $\mathfrak{su}(N)$ Lie algebra. The derivation is based on a relation linking the index of a generator to the indexes of its non-null elements. The closed formulas obtained to compute the values of the structure constants are simple expressions involving those indexes and can be analytically evaluated without any need of the expression of the generators. We hope that these expressions can be widely used for analytical and computational interest in Physics.