[Notes] Introduction to Lie Algebra

Textbook: Introduction to Lie Algebra Erdmann & Wildon

這是林紹雄老師開李代數的課本，我沒有上他的課，不過我自己拿這本書來唸。這本書很新，2005年才出版，我認為非常適合入門，章節分段適中，習題難度恰當(我幾乎全消了 不過解答沒有掃描就是了 xD)，核心是在介紹半單李代數(semisimple Lie algebra)的結構，非常美妙。

不過無限維李代數和李代數和李群之間的關係這本書變沒有提及，等我之後唸到了再補上。


Notes
Introduction to Lie Algebra
ideals, homomorphisms, derivarions, commutator algebras,
quotient algebras, Isomorphism Theorems
Low dimensional Lie Algebras
Heisenberg Algebra, $sl(2,\mathbf{C})$

Notes
Solvable/Semisimple/Nilpotent Lie Algebras
Subalgebras of $gl(V)$
weights, Invariance Lemma, Engel's Thm, Lie's Thm
Lie Algebra Representations
faithful rep., Lie Module, Irreducible Modules, Isomorphism Theorems
Schur Lemma, Ado's Thm
$sl(2,\mathbf{C})$-module representation

Notes
Finite irreducible $sl(2,\mathbf{C})$-modules
highest weight vector
Jordan Decomposition
Abstract Jordan Decomposition
existence and uniqueness
Casimir Operator
Weyl Theorem
Cartan Criteria
Bilinear Form
Killing Form

Notes
Cartan Criteria
Finite Complex Semisimple Lie Algebras
Root Space Decomposition
Weight spaces under adH
Cartan's Subalgebra/ Maximal Toral Subalgebra
$sl(\alpha)$-module and eigenvalues
Root Strings
Chevalley Theorem
Cartan's Subalgebra as Inner product
Reflections

Notes
Root Systems
Finiteness Lemma
Irreducible Root System
Base-existence Theorem
Simple Roots/Reflections
Weyl Group
Cartan Matrix and Dynkin Diagram
Root Space Isomorphisms
Dual Root System

Notes
Classical Lie Algebras
$sl( l+1, \mathbf{C}) : A_l$
$so( 2l+1, \mathbf{C}) : B_l$
$sl( 2l, \mathbf{C}) : D_l$
$sp( 2l, \mathbf{C}) : C_l$
Simplicity
Killing Form
Isomorphisms
Classification of Dynkin Diagrams
Admissible Subsets
Shrinking Lemma
Exceptional Dynkin Diagrams
Type $G_2$, $F_4$, $E_6$, $E_7$, $E_8$
Complex semisimple Lie Algebras
Lie Algebra Generators
Serre's Relation/Theorem