Z 分次表示理论

箭图Hecke 代数的Z 分次表示理论是“代数群、量子群及Hecke 代数” 领域中当前最活跃的研究方向之一. 箭图Hecke 代数及其分圆版本产生于对量子群及其可积最高权表示的范畴化的研究, 它们与数学及数学物理的许多不同分支如Lie 代数、量子群、Kazhdan-Lusztig 理论、代数几何(箭图簇,反常层)、扭结理论、拓扑量子场论(TQFT) 等都有着紧密的联系与相互作用. 本文详细介绍了该方向的最新进展、前沿以及研究前景.

The Z-graded representation theory

Currently the Z-graded representation theory of quiver Hecke algebras is one of the most active research topics in the area of algebraic groups,quantum groups and Hecke algebras.Quiver Hecke algebras and their cyclotomic versions arise from the study of categori？cation of quantum groups and their integrable highest weight modules.They have intimate connection and interactions with many different branches in mathematics and mathematical physics like Lie algebras,quantum groups,Kazhdan-Lusztig theory,algebraic geometry（quiver variety,perverse sheaves）,knot theory,topological quantum ？eld theory（TQFT）,etc.In this article,we give a survey of the latest development,research frontier and future prospect in this area.