Linking the special orthogonal algebra so4 and the tetrahedron algebra

• Main Author: Morales, John Vincent S.
• Format: text
• Published: Animo Repository 2022
• Subjects: Combinatorial designs and configurations; Algebras, Linear; Polynomials; Algebra
• Online Access: https://animorepository.dlsu.edu.ph/faculty_research/11538
• Institution: De La Salle University

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 : x → so4 which can be extended to a homomorphism φ : x → 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].