The equitable basis for ${\mathfrak{sl}_2}$

Abstract

This article contains an investigation of the equitable basis for the Lie algebra ${\mathfrak{sl}_2}$. Denoting this basis by {x, y, z}, we have

$$[x,y] = 2x + 2y, \quad [y,z] = 2y + 2z, \quad [z, x] = 2z + 2x.$$

We determine the group of automorphisms G generated by exp(ad x*), exp(ad y*), exp(ad z*), where {x*, y*, z*} is the basis for ${\mathfrak{sl}_2}$ dual to {x, y, z} with respect to the trace form (u, v) = tr(uv) and study the relationship of G to the isometries of the lattices ${L={\mathbb Z}x \oplus {\mathbb Z}y\oplus {\mathbb Z}z}$ and ${L^* ={\mathbb Z}x^*\oplus {\mathbb Z}y^*\oplus {\mathbb Z}z^*}$. The matrix of the trace form is a Cartan matrix of hyperbolic type, and we identify the equitable basis with a set of simple roots of the corresponding Kac–Moody Lie algebra ${\mathfrak{g}}$, so that L is the root lattice and ${\frac{1}{2} L^*}$ is the weight lattice of ${\mathfrak g}$. The orbit G(x) of x coincides with the set of real roots of ${\mathfrak g}$. We determine the isotropic roots of ${\mathfrak g}$ and show that each isotropic root has multiplicity 1. We describe the finite-dimensional ${\mathfrak{sl}_2}$-modules from the point of view of the equitable basis. In the final section, we establish a connection between the Weyl group orbit of the fundamental weights of ${\mathfrak{g}}$ and Pythagorean triples.