胞腔代数和仿射胞腔代数简介

表示论中一个最基本的问题是确定不可约表示的参数集,这个问题至今没有完全解决.对于Graham和Lehrer引入的有限维胞腔代数,这个问题得到了完满解答,并被成功地应用于数学和物理中出现的许多代数.近来,人们引入仿射胞腔代数,将Graham和Lehrer有限维胞腔代数的表示理论框架推广到一类无限维代数上.仿射胞腔代数不仅包括有限维胞腔代数,也包括无限维的仿射Temperley-Lieb代数和Lusztig的A-型仿射Hecke代数.本文将对胞腔代数的发展历史和主要研究成果做一些综述,同时,对新引入的仿射胞腔代数及其最新成果做一点简介.

Cellular and Affine Cellular Algebras

Cellular algebras in the sense of Graham and Lehrer provide a beautiful model to understand irreducible representations by linear algebra.In this note,we survey first this theory of cellular algebras developed in the last decade,and then introduce the so-called affine cellular algebras,which include the infinite dimensional affine Hecke algebras of type A,and outline some new results on affine cellular algebras.In particular,we apply the new theory to affine Hecke algebras of type A,and get when these algeb...