I will review the existing constructions of various tensor category structures on module categories for affine Lie algebras. I will discuss the results that were first conjectured in the work of Moore and Seiberg and that led us to the construction of the modular tensor category structure in the positive integral level case. Then I will review the existing constructions and results in the following three cases: (i) the level plus the dual Coxeter number is not a nonnegative rational number, (ii) the level is a positive integer and (iii) the level is an admissible number. I will also present several open problems. I will give historical remarks in the talk to correct many misunderstandings and confusions existed in the literature.

黄一知，美國罗格斯大学数学系教授。数学杂志Communications in Contemporary Mathematics共同主编，数学杂志New York Journal of Mathematics编委会成员。研究量子场论的数学理论及其在代数、拓扑、几何、凝聚态物理和弦论中的应用。复旦大学数学系力学专业（航空航天系前身)学士，复旦大学数学研究所硕士，罗格斯大学博士。