Lectures on Lie algebras and related topics
10:00am-12:00pm
Xuanzhong Dai (Fudan University)
Lecture 1: Semisimple Lie algebras
In this lecture, we will study a semisimple Lie algebra L over complex field C via its adjoint representation. We will see that L is built up from sl(2, C), so we first study representation of sl(2,C). Then we introduce Weyl’s Theorem on complete reduciblity, which is a fundamental theorem in the representation theory of semisimple Lie algebras.
Lecture 2: Root system
In this lecture, we will discuss about the root systems. A root system is considered as a subset in the Euclidean space satisfying certain geometrical properties. We will classify the irreducible root systems or equivalently the connected Dynkin diagrams, which finally leads to a construction of simple Lie algebras of type A-G.
Lecture 3: Universal enveloping algebras
In this lecture, we will introduce the construction of universal enveloping algebra. The ideal of Universal enveloping algebra is to embed any Lie algebra L to an associated algebra U(L) with the identity such that the bracket of elements in L coincides with the usual commutation. Then we will introduce the Poincare-Birkhoff-Witt theorem (PBW theorem), which gives a description of universal enveloping algebra.
Lecture 4: Character formulas and Harish-Chandra theorem
In this lecture, we will define the formal character, namely the formal sum of weights of representations of L. We will introduce a theorem of Harish-Chandra, which identifies the center of the universal enveloping algebra and the symmetric algebra of a Cartan subalgebra that is invariant under the Weyl group. Then we apply Harish-Chandra theorem to obtain some remarkable formulas for the characters and multiplicities of finite dimensional L modules.
Lecture 5: Chiral de Rham complex
In this final lecture, we will introduce some recent progress about the chiral de Rham complex and chiral differential operators on the upper half plane. For any congruence subgroup Gamma, we consider the vertex operator algebra constructed from Gamma-invariant global sections of the chiral de Rham complex on the upper half plane which are holomorphic at all the cusps. We will see that the vertex operations can be expressed in terms of modified Rankin-Cohen brackets of modular forms.