距离正则图是代数组合论的核心课题。以三对角对、勒纳德对和勒纳德三元组为主要研究内容的理论是近年兴起的研究距离正则图的新理论。这些新理论统称为Terwilliger代数的表示理论。这些新理论给距离正则图的研究带来了全新的方法，受到人们越来越多的关注。本项目研究：（1）三对角对和勒纳德三元组的分类；（2）利用有限典型群几何、Terwilliger代数和与之相关的李代数、量子群，构作三对角对、勒纳德对和勒纳德三元组；（3）研究三对角对、勒纳德对、勒纳德三元组在距离正则图上的应用。这些新方法和理论的研究，不仅对代数组合的研究有重要意义，而且对典型群、李代数和量子群的研究也有重要意义。

Distance regular graphs are a key topic in algebraic combinatorics. The theory whose main contents are tridiagonal pairs, Leonard pairs and Leonard triples is a new one formed in recent years for studying distance regular graphs. These new theories are known collectively as the representations of the Terwilliger algebras. They bring new methods on studying distance regular graphs. More and more people are interested in it. In this research, we study (1) the classification of tridiagonal pairs and Leonard triples; (2) the constructions of tridiagonal pairs, Leonard pairs and Leonard triples using geometry of classical groups over finite fields,Terwilliger algebras and the related Lie algebras and quantum groups; (3) the applications of tridiagonal pairs, Leonard pairs and Leonard triples to distance regular graphs. The study on these new methods and theories make an important role not only in algebraic combinatorics, but also in classical groups, Lie algebras and quantum groups.