Philippine E-Journals

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 x, called the tetrahedron algebra, using generators {xij | i,j  ∈  {1,2,3,4}, I ≠ 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 distance regular 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 Ai = 1= Ai (x) denote the dual adjacency matrix of S. Let T = T(x) denote the subalgebra of MatX(C) generated by A1 and A1. In this paper, we exhibit an action of [1] 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*,  in T by giving the matrix representations of the restriction on Ul with respect to the basis Bi. Finally, we take A* +       +B* ô€€€ , A + B ô€€€ , K ô€€€      and K* ô€€€   , and show that these matrices satisfy the relations of  x.