The isotropic harmonic oscillator

While studying Lie-algebras I read that the three-dimensional harmonic oscillator has an $SU(3)$ symmetry. I found this very unexpected; I thought it was ''obvious'' that the symmetry is only $SO(3)$.

$F_4$ tensor products

The product of two irreducible representations of a simple Lie algebra can be decomposed into irreducible components. There are various techniques to calculate this decomposition, see for example chapter XIV in [1]. However, the decomposition can also be calculated by brute force.

Weight diagrams of $G_2$

This post contains pictures of weight diagrams of irreducible representations of the Lie algebra $G_2$.

The 7-dimensional representation of $G_2$

The exceptional Lie algebra $G_2$ has an irreducible representation of dimension 7. In this post I calculate the matrices of this irrep.

Commutation relations in $G_2$

In this post I calculate the structure constants of the exceptional Lie algebra $G_2$. I assume the reader is familiar with Lie algebras, for example at the level of chapter 9 in [1].