Representations of TerwiUiger algebras and their applications

本研究项目的主要成果（将在后面详细讨论）是在P-和Q-多项式型Terwilliger代数的非薄表示领域取得了突破。其中一些开创性的工作是关于环切方案的Terwilliger代数和Jacobi和，关于自旋模型，量子群和Terwilliger代数之间的关系，关于II型矩阵的结构。设T为具有经典参数的P-和q -多项式型Terwilliger代数，其中经典参数是指比通常少一个变量的代数。我们得到了以下结果。端点1的不可约t模具有阶梯基（Hobart-Ito）。在最简单的参数情况下，不可约t模是通过On sager代数（Ito-Tanabe-Terwilliger）的有限维不可约表示确定的。得到了t模结构的一个基本定理，使我们能够通过定义一个On -On - sager代数的q-类似（q-On - sager代数）（Ito-Tanabe-Terwilliger）来处理一般情况。因此，在经典参数的情况下，不可约t模的问题被简化为确定q-Onsager代数的有限维不可约表示。如果直径为3，则通过仿射量子代数U_q （sl_2）的类型（1,1）表示确定了q-0n个sager代数的有限维不可约表示。这是一开始提到的突破，我们的目标是将其推广到任意直径。

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.