A Generalization of the Terwilliger Algebra

P. M. Terwilliger (1992, J. Algebraic Combin.1, 363–388) considered the C-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 determine much of the structure of T, thought not all of it in general. To illuminate the role these relations play, we consider a certain generalization T of T. To go from T to T, we replace M and M* with a pair of dual character algebras C and C*. We define T by generators and relations; intuitively T is the associative C-algebra with identity generated by C and C* subject to the analogues of Terwilliger's relations. T is infinite dimensional and noncommutative in general. We construct an irreducible T-module which we call the primary module; the dimension of this module is the same as that of C and C*. We find two bases of the primary module; one diagonalizes C and the other diagonalizes C*. We compute the action of the generators of T on these bases. We show T is a direct sum of two sided ideals T0 and T1 with T0 isomorphic to a full matrix algebra. We show that the irreducible module associated with T0 is isomorphic to the primary module. We compute the central primitive idempotent of T associated with T0 in terms of the generators of T.