July 6-July 13，2019

（青岛全国李代数会议预备学校）

主办单位：华东师范大学数学科学学院

2019年华东师大李理论暑期学校（青岛全国李代数会议预备学校（*））于2019年7月6日至7月13日在华东师范大学闵行校区举行。暑期学校开设四门课程，每门课程包括6小时讲授与2小时答疑。 课程内容为：

课程（1）：“Real Reductive Lie Groups and Discrete Series Representations”（讲授人：黄劲松教授， The Hong Kong University of Science and Technology）；

课程（2）：“Yangians of Classical Lie Algebras”（讲授人：景乃桓教授 , North Carolina State University）；

课程（3）：“Nilpotent Orbits, Primitive Ideals and Finite W-algebras”（讲授人：Alexander Premet教授，The University of Manchester）；

课程（4）：“Quantum Groups and Quiver Representations”（讲授人：王伟强教授，University of Virginia）。

欢迎各高校的研究生与青年教师参加（下载注册表）。这次暑期学校得到了华东师范大学研究生院以及数学科学学院的大力支持。

暑期学校组委会：董崇英（UC, Santa Cruz）

舒 斌（华东师范大学）

王伟强（University of Virginia）

王宪栋（青岛大学）

暑期学校联系人：孙 东（华东师范大学 18701811739 邮箱：1977348054@qq.com）

薛琛亮（华东师范大学）

张红艳（华东师范大学）

（*）青岛全国李代数会议将于2019年7月14日至7月20日在青岛大学召开。会议主页为：http://liealgebra2019.csp.escience.cn/dct/page/1

黄劲松教授,The Hong Kong University of Science and Technology；

景乃桓教授, North Carolina State University；

Alexander Premet教授，The University of Manchester；

王伟强教授， University of Virginia。

（1）Real Reductive Lie Groups and Discrete Series Representations by Professor Jing-Song Huang

**Abstract:** The classification of discrete series representations by Harish-Chandra in 1960's was one of the greatest achievements of mathematics in 20th century. It laid down the foundation for the Langlands classification of admissible representations and the Langlands program. This mini-course is an introduction to Harish-Chandra theory of discrete series by using more recently developed techniques. We will begin with Schur's orthogonality relations for irreducible representations of compact Lie groups. The natural generalization is Harish-Chandra's orthogonality relations for discrete series of noncompact reductive Lie groups. We will also discuss the further generalization of Harish-Chandra's orthogonality relations to admissible representations.

（2）Yangians of classical Lie algebras by Professor Naihuan Jing

**Abstract:** The Yangian algebra was introduced by Drinfeld as a nontrivial
example of noncommutative and noncocomutative Hopf algebra generalization
of the enveloping algebra of a simple Lie algebra. First of part of the
lectures will be devoted the Yangian of the general linear Lie algebra (type A). We
will start with the first presentation using the R-matrix and go over some of its fundamental
properties, then we will discuss the other two presentations and their identification. In
the second part we will introduce the twisted Yangians as special subalgebras of the Yangian in type A to realize
other types (BCD types) of Yangians and their presentations and possibly some basics of their
representations.

（3）Nilpotent orbits, primitive ideals and finite W-algebras by Professor Alexander Premet.

**Abstract:** I will discuss several topics of modular representation theory of restricted Lie algebras.

This will include the following topics:

1. reduced enveloping algebras and the current state of the first Kac-Weisfeiler conjecture;

2. nilpotent orbits and sheets in reductive Lie algebras;

3. support varieties for non-restricted modules and the second Kac-Weisfeiler conjecture;

4. finite W-algebras and their modular analogues;

5. small modular representations and multiplicity-free primitive ideals;

6. some open problems.

（4）Quantum groups and quiver representations by Professor Weiqiang Wang

**Abstract:** The goal of this mini-course is to provide an introduction to some basic structures of quantum groups from the viewpoint of quiver representations.
Quantum groups are introduced by Drinfeld and Jimbo in 1980's as a deformation of the universal enveloping algebras of simple Lie algebras. A quantum group admits automorphisms (due to Lusztig) which satisfy the braid group relations for the associated Weyl group.

Dynkin quivers are ADE Dynkin diagrams equipped with orientations. In this mini-course we study representations of a Dynkin quiver (equivalently, representations of a so-called path algebra), and classify its indecomposables via positive roots of the corresponding simple Lie algebra. By studying the quiver representations over finite fields, we formulate the notion of Ringel-Hall algebra and show that it provides a realization of a positive half of a quantum group, U^+. The reflection functors are formulated, which provide a construction of the braid group action on the quantum group. We shall describe PBW bases for U^+ (as well as monomial bases, and canonical basis, if time permitting).

7/Sun | 8/Mon | 9/Tue | 10/Wen | 11/Thu | 12/Fri | 13/Sat | |

(Break of Summer School) | |||||||

University Lectures(Colloquium talks) | |||||||

Venue | Lecture Hall 401, Math Building 数学楼401报告厅 |
Classroom 310，Teachin Building 3 三教310教室 |
Classroom 310，Teachin Building 3 三教310教室 |
Lecture Hall 102，Math Building 数学楼102报告厅 |
Classroom 310，Teachin Building 3 三教310教室 |
Classroom 310，Teachin Building 3 三教310教室 |
Classroom 310，Teachin Building 3 三教310教室 |

Morning 9:00-10:00 |
Jing | Jing | Wang | Premet (9:00-10:00 am) |
Huang | Huang | Huang |

Break time 10:00-10:20 |
Jing (10:10-11:10am) | ||||||

10:20-11:20 | Jing | Jing | Wang | Huang | Huang | Huang | |

Noon break | |||||||

Afternoon 2:00-3:00 |
Wang | Premet | Jing | Huang(2:00-3:00 pm) | Premet | Wang | Premet |

Break time 3:00-3:20 |
Wang(3:10-4:10 pm) | ||||||

3:20-4:20 | Wang | Premet | Jing | Premet | Wang | Premet | |

Answer to Questions 4:20-5:20 |

Coming soon...