Abstract :
Terwilliger [J. Algebraic Combin.1 (1992), 363–388] considered the -algebra generated by a given Bose Mesner algebra M and the associated dual Bose Mesner algebra M*. This algebra is now known as the Terwilliger algebra and is usually denoted by T. Terwilliger showed that each vanishing intersection number and Krein parameter of M gives rise to a relation on certain generators of T. These relations are often called the triple product relations. They determine much of the structure of T, though not all of it in general. To illuminate the role these relations play, the current author introduced [J. Algebra233 (2000), 213–252] a generalization of T. To go from T to , we replace M and M* with a pair of dual character algebras C and C*. The dimensions of C and C* are equal; let d + 1 denote this common dimension. Intuitively, is the associative -algebra with identity generated by C and C* subject to the analogues of Terwilligerʹs triple product relations. is infinite-dimensional and noncommutative in general. In this paper we study and its finite-dimensional modules when d = 2 and has no “extra” vanishing intersection numbers or dual intersection numbers. In this case we show is -algebra isomorphic to M3( ) , where M3( ) denotes the -algebra consisting of all 3-by-3 matrices with entries in and denotes the associative -algebra with identity generated by the symbols e and f subject to the relations e2 = e and f2 = f. We find a basis for and we determine the center of . We classify the finite-dimensional indecomposable -modules up to isomorphism. There are four such -modules in every odd dimension, and in every even dimension these modules are parameterized by a single complex number. We also classify the finite-dimensional irreducible -modules up to isomorphism. Using our results concerning , we find a basis for , we describe the center of , and we classify both the finite-dimensional indecomposable and the finite-dimensional irreducible -modules up to isomorphism.
