The Equitable Basis for sl_2

TL;DR

This paper explores the equitable basis for sl_2, revealing its automorphism group as PSL_2(Z), relating it to Kac-Moody algebras, root lattices, and connecting Weyl group orbits with Pythagorean triples.

Contribution

It establishes the structure of the automorphism group for the equitable basis of sl_2 and links it to hyperbolic Kac-Moody algebras and Pythagorean triples, providing new geometric insights.

Findings

Automorphism group G is isomorphic to PSL_2(Z)

Orbit G(x) equals {u in L | (u,u)=2}

Weyl group orbit of weights relates to Pythagorean triples

Abstract

This article contains an investigation of the equitable basis for the Lie algebra sl_2. Denoting this basis by {x,y,z}, we have [x,y] = 2x + 2y, [y,z] = 2y + 2z, [z, x] = 2z + 2x. One focus of our study is the group of automorphisms G generated by exp(ad x*), exp(ad y*), exp(ad z*), where {x*,y*,z*} is the basis for sl_2 dual to {x,y,z} with respect to the trace form (u,v) = tr(uv). We show that G is isomorphic to the modular group PSL_2(Z). Another focus of our investigation is the lattice L=Zx+Zy+Zz. We prove that the orbit G(x) equals {u in L |(u,u)=2}. We determine the precise relationship between (i) the group G, (ii) the group of automorphisms for sl_2 that preserve L, (iii) the group of automorphisms and antiautomorphisms for sl_2 that preserve L, and (iv) the group of isometries for (,) that preserve L. We obtain analogous results for the lattice L* =Zx*+Zy*+Zz*. Relative to the equitable basis, the matrix of the trace form is a Cartan matrix of hyperbolic type; consequently,we identify the equitable basis with the set of simple roots of the corresponding Kac-Moody Lie algebra g. Then L is the root lattice for g and 1/2L* is the weight lattice, and G(x) coincides with the set of real roots for g. Using L, L*, and G, we give several descriptions of the isotropic roots for g and show that each isotropic root has multiplicity 1. We describe the finite-dimensional 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 g and Pythagorean triples.