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.