The Terwilliger algebra of digraphs I -- Hamming digraph $H^{\ast }(d,3)$

In the present paper, we define the Terwilliger algebra of digraphs. Then, we determine the irreducible modules of the Terwilliger algebra of a Hamming digraph $H^{\ast }(d,3)$. As is well known, the representation of the Terwilliger algebra of a binary Hamming graph $H(d,2)$ is closely related to that of the Lie algebra ${sl}_2(\mathbb{C})$. We show that in the case of $H^{\ast }(d,3)$, it is related to that of the Lie algebra ${sl}_3(\mathbb{C})$. We also identify the Terwilliger algebra of $H^{\ast }(d,3)$ as the $d$ symmetric tensor algebra of ${\mathrm{M}\mathrm{a}\mathrm{t}}_3(\mathbb{C})$.