caa a22 4500 445835885 CHVBK 20180317145330.0 cr unu---uuuuu 170323e20110601xx s 000 0 eng 10.1007/s00209-010-0682-9 doi (NATIONALLICENCE)springer-10.1007/s00209-010-0682-9 The equitable basis for $${\mathfrak{sl}_2}$$ [Elektronische Daten] [Georgia Benkart, Paul Terwilliger] 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. Springer-Verlag, 2010 Equitable basis nationallicence Modular group nationallicence Kac-Moody algebra nationallicence Cartan matrix nationallicence Benkart Georgia Department of Mathematics, University of Wisconsin, 53706, Madison, WI, USA aut Terwilliger Paul Department of Mathematics, University of Wisconsin, 53706, Madison, WI, USA aut Mathematische Zeitschrift Springer-Verlag 268/1-2(2011-06-01), 535-557 0025-5874 268:1-2<535 2011 268 209 https://doi.org/10.1007/s00209-010-0682-9 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00209-010-0682-9 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Benkart Georgia Department of Mathematics, University of Wisconsin, 53706, Madison, WI, USA aut NATIONALLICENCE 700 1- Terwilliger Paul Department of Mathematics, University of Wisconsin, 53706, Madison, WI, USA aut NATIONALLICENCE 773 0- Mathematische Zeitschrift Springer-Verlag 268/1-2(2011-06-01), 535-557 0025-5874 268:1-2<535 2011 268 209 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer