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- Title
The equitable basis for $${\mathfrak{sl}_2}$$.
- Authors
Benkart, Georgia; Terwilliger, Paul
- 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 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.
- Subjects
LIE algebras; AUTOMORPHISMS; NUMERICAL roots; WEYL groups; LINEAR algebra
- Publication
Mathematische Zeitschrift, 2011, Vol 268, Issue 1/2, p535
- ISSN
0025-5874
- Publication type
Article
- DOI
10.1007/s00209-010-0682-9