The Tetrahedron algebra, the Onsager algebra, and the ${\mathfrak{sl}}_2$ loop algebra

Abstract

Let $K$ denote a field with characteristic 0 and let $T$ denote an indeterminate. We give a presentation for the three-point loop algebra ${\mathfrak{sl}}_2\otimes K[T,T^{-1},(T-1{)}^{-1}]$ via generators and relations. This presentation displays $S_4$-symmetry. Using this presentation we obtain a decomposition of the above loop algebra into a direct sum of three subalgebras, each of which is isomorphic to the Onsager algebra.

25 pages