Linking the special orthogonal algebra [formula omitted] and the tetrahedron algebra ⊠

Abstract

In 2007, B. Hartwig and Terwilliger found a presentation for the three-point sl2 loop algebra in terms of generators and relations. To obtain this presentation, they defined a Lie algebra ⊠ by generators and relations and established an isomorphism from ⊠ to three-point sl2 loop algebra. Essentially, ⊠ has six generators which can be naturally identified with the six edges of the tetrahedron. In fact, each face of the tetrahedron has three surrounding edges which generate a subalgebra of ⊠ that is isomorphic to sl2. It is interesting to know whether a direct sum of finitely many copies of sl2 (e.g., special orthogonal algebra so4) captures the bracket relations of the generators of ⊠. Here, we show that there exists a Lie algebra homomorphism ϕ:⊠→so4 which can be extended to a homomorphism ϕ:⊠→L where L is a direct sum of finitely many copies of sl2. We construct a finite-dimensional so4-module which is viewed as a ⊠-module via the homomorphism ϕ. We show how this so4-module is related to Krawtchouk polynomials. This paper is inspired by and is an extension of the work of Nomura and Terwilliger (2012) [19].