The Tetrahedron Algebra and Shrikhande Graph
J. Morales (pp. 10-18)
Abstract
Hartwig and Terwilliger (2007) obtained a presentation of the three-point sl2 loop algebra via generators and relations. In order to do this, they defined a complex Lie algebra , called the tetrahedron algebra, using generators {xij | i, j ∈ {1, 2, 3, 4}, i 6= j} and relations: (i) xij + xji = 0, (ii) [xhi, xij ] = 2xhi + 2xij for mutually distinct h, i, j and (iii) [xhi, [xhi, [xhi, xjk]]] = 4[xhi, xjk] for mutually distinct h, i, j, k. The Shrikhande graph S was introduced by S. S. Shrikhande in 1959. Egawa showed that S is a distanceregular graph whose parameters coincide with that of the Hamming graph H(2, 4). Let X be the vertex set of S. Let A1 denote the adjacency matrix of S. Fix x ∈ X and let A∗ 1 = A∗ 1 (x) denote the dual adjacency matrix of S. Let T = T(x) denote the subalgebra of M atX(C) generated by A1 and A∗ 1. In this paper, we exhibit an action of on the standard module of S. To do this, we use the complete set of pairwise nonisomorphic irreducible T−modules Ul’s of S and the standard basis Bl of each Ul which were obtained by Tanabe in 1997. We define matrices A, A ∗ , B, B ∗ , K, K ∗ , Φ and Ψ in T by giving the matrix representations of the restriction on Ul with respect to the basis Bi . Finally, we take A ∗ + Ψ + Φ, B ∗ − Φ, A − Ψ + Φ, B − Φ, K − Ψ and K ∗ − Ψ, and show that these matrices satisfy the relations of