Introduction

This is a student seminar in USC in Fall 2022. Our aim is to understand the classification and representation theory of compact Lie groups. We'll go through the first four chapters of [Bröcker-tom Dieck] and possibly the last chapter if we had time. All undergrad and graduate students at USC are welcomed to participate in, and participants are encouraged to give a talk on some of the contents. The minimal prerequisites for this seminar are algebra(MATH 410) and real analysis(MATH 425ab). For those interested in the seminar, please email liusiyan@usc.edu so that I can add you to the mailing list. We'll meet at 1:00-2:20pm every Friday.

Recordings&Notes

• Introduction - by Siyang Liu

• Definitions and examples; left-invariant vector fields and one-parameter groups; exponential map - by Boxi Hao

• Representations; semisimple modules; operations of representations; characters and orthogonality relations - by Jishnu Bose

• Representations of SU(2), SO(3), U(2) and O(3); real and quaternion representations; the character ring and the representation ring - by Linds Wise

• Representations of abelian groups; Representations and Lie algebras; the Lie algebra sl(2,C) - by Sung Kim

• Peter-Weyl theorem - by Haosen Wu

• Group algebras; induced representation; complexification of Lie groups - by Si-Yang Liu

• Maximal tori and Weyl groups - by Robin Rong

• Root Systems and Dynkin Diagrams - by Fan Yang

References

1. [Bröcker-tom Dieck] Bröcker, T., & Dieck, T. (1985). Representations of Compact Lie Groups. Springer.