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

Brian Hartwig·Paul Terwilliger
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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 $\mathfrak{sl}_2 \otimes K\lbrack T, T^{-1},(T-1)^{-1}\rbrack$ 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.

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Brian Hartwig, Paul Terwilliger. 2005-11-02. The Tetrahedron algebra, the Onsager algebra, and the $\mathfrak{sl}_2$ loop algebra. https://arxiv.org/abs/math-ph/0511004

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