Federico Binda and Ziyang Gao
will present two seminars on Lie Algebras – Representation theory
Here is a small note of the seminar.
Binda’s abstract:
Root systems provide a relatively uncomplicated way of completely characterizing simple and semisimple Lie algebras. We follow the axiomatic approach (as in Serre [2], Humphreys [1]). We introduce bases, the Weyl group and we explain its action on the set of bases (or, equivalently, on the Weyl chambers). Finally we introduce the classification theorem, using the Cartan matrix, Coxeter graphs and Dynkin diagrams.
References:
[1] J. Humphreys (1972), Introduction to Lie Algebras and Representation Theory, GTM, Springer-Verlag, NY.
[2] J. P. Serre (1966), Algébres de Lie semi-simples complexes, W.A. Benjamin, NY.
Gao’s abstract:
In this lecture, our main subject is the root space decomposition, a useful method to describe the representations of a Lie algebra. We will mainly focus on the case of 
. We will first characterize all irreducible representations of 
in terms of highest weight, then study the general root space decomposition. The notion of root system will be introduced here. Finally, we will generalize this notion and give the definition and a few basic properties of abstract root system.
References:
[1] J.Humphreys(1972), Introduction to Lie Algebras and Representation Theory, GTM-9, Springer-Verlag, NY.
[2] Jinpeng An, Lecture Notes, on-line version: ftp://162.105.69.120/teachers/anjp/李群与齐性空间