Representations of TerwiUiger algebras and their applications

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The major outcome of this research project, which will be discussed in detail later, is that there was a breakthrough in the area of the non thin representations of Terwilliger algebras of P- and Q- polynomial type. Among others are some pioneering works on the Terwilliger algebras of cyclotomic schemes and the Jacobi sums, on relations between spin models, quantum groups and Terwilliger algebras, on the structure of type II matrices. Let T be a Terwilliger algebra of P- and Q-polynomial type with classical parameters, where the classical parameters mean the ones with one less variables than usual. We obtained the following results.The irreducible T-modules of endpoint 1 have a ladder basis (Hobart-Ito). In the simplest case of parameters, irreducible T-modules are determined via finite dimensional irreducible representations of On sager algebras (Ito-Tanabe-Terwilliger). A basic theorem is obtained for the structure of T-modules, enabling us to deal with the general case by defining the q-analogue of an On sager algebra (q-On sager algebra) (Ito-Tanabe-Terwilliger).Thus in the case of classical parameters, the problem of irreducible T-modules is reduced to the determination of finite dimensional irreducible representations of q-Onsager algebras. If the diameter is 3, finite dimensional irreducible representations of q-0n sager algebras are determined via the type (1,1) representations of the affine quantum algebra U_q (sl_2). This is the breakthrough mentioned at the beginning and we are aiming at generalizing it to arbitrary diameters.