[Notes] Representations of Semisimple Lie Algebras in the Category O

 Author: Humphreys 

 在程老師欽點下開始念這本書，Humphreys 的書以難讀出名，於是我試圖將這本書裡面的敘述寫成較有系統的形式並且補完一些預備知識，

目前進度: Chp 2
持續更新

Chapter 0: Review of Semisimple Lie Algebras
Cartan Decomposition
Root System
Weyl Groups
Chevalley-Bruhat Ordering of $W$
Universal Enveloping Algebra
Integral Weights
Representations
Finite Dimensional Modules
Simple Modules for $\mathfrak{sl}(2,\mathbb{C})$


Chapter 1: Catergory O: Basics
Revivew on Category Theory
Axioms and Consequences
Highest Weight Modules
Verma Modules and Simple Modules
Maximal Vectors in Verma Modules
Example: $\mathfrak{sl}(2,\mathbb{C})$
Finite Dimensional Modules
Action of the Center
Central Characters and Linked Weights
Harish-Chandra Homomorphism
Harish-Chandra Theorem
Category $\mathcal{O}$ is Artinian
Subcategories $\mathcal{O}_\chi$
Blocks
Formal Characters of Finite Dimensional Modules
Formal Characters of Modules in $\mathcal{O}$
Formal Characters of Verma Modules

Chapter 2: Characters of Finite Dimensional Modules
Summary of Prerequisites
Formal Characters Revisited
The Functions $p$ and $q$
Formulas of Weyl and Kostant
Dimension Formula
Maximal Submodule of $M(\lambda):\lambda\in\Lambda^+$

Chapter 3: Category O: Methods
Hom and Ext

Duality in $\mathcal{O}$

Duals of Highest Weight Modules

The Reflection Group $W_{[\lambda]}$

Dominant and Antidominant Weights

Tensoring Verma Modules with Finite Dimensional Modules

Standard Filtrations

Projectives in $\mathcal{O}$

Indecomposable Projectives

Standard Filtrations of Projectives

BGG Reciprocity

Example: $\mathfrak{sl}(2,\mathbb{C})$

Projective Generators and Finite Dimensional Algebras

Contravariant Forms

Universal Construction