#1: February 4

Reference: [1] Sec. 1 & 2

Contents
Lie algebra •
$\mathfrak{gl}(V)$, Abelian Lie algebras •
Lie subalgebra •
linear Lie algebras •
structure constants •
homo-, mono-, epi-, isomorphism •
ideal •
simple Lie algebra, $\mathfrak{sl}(2,\mathbb{F}),\mathfrak{sl}(V)$ •
derivations: inner ($\mathrm{ad}x$), outer •
automorphisms, nilpotent derivation •
classical Lie algebras: $A_\ell=\mathfrak{sl}(\ell+1,\mathbb{F})$,
$B_\ell=\mathfrak{o}(2\ell+1,\mathbb{F})$, $C_\ell=\mathfrak{sp}(2\ell,\mathbb{F})$,
$D_\ell=\mathfrak{o}(2\ell,\mathbb{F})$.

#2: February 11

Reference: [1] Sec. 3

Contents
derived series •
solvable Lie algebra •
$\mathfrak{t}(n,\mathbb{F})$ upper triangular matrices •
radical (maximal solvable ideal) •
semisimple Lie algebra •
nilpotent Lie algebra •
Engel's theorem •
flag.

#3: February 18

Reference: [1] Sec. 4

Contents
Lie's theorem •
Jordan-Chevalley decomposition.

#4: February 25

Reference: [1] Sec. 4 & 5

Contents
Cartan's criterion for solvability •
Cartan's criterion for semisimplicity •
Killing form ($\kappa$) •
abstract Jordan-Chevalley decomposition.

#5: March 4

Reference: [1] Sec. 6

Contents
$L$-modules •
Schur's lemma •
Casimir operator •
Weyl's theorem.

#6: March 11

Reference: [1] Sec. 7 & 8

Contents
weights and weight spaces •
maximal vector •
heighest weight •
representations of $\mathfrak{sl}(2,\mathbb{F})$ •
toral subalgebra •
roots •
rootspace decomposition.

#7: March 18

Reference: [1] Sec. 8

Contents
orthogonal properties •
Cartan integers •
properties of root systems.

#8: March 25

Reference: [1] Sec. 9

Contents
reflections in a Euclidean space •
abstract root systems •
low-rank examples •
pairs of roots •
$\alpha$-string through $\beta$.

#9: April 1

Reference: [1] Sec. 10

Contents
base of a root system (simple roots) •
height of a root, positive & negative roots •
$(\alpha,\beta)\leq 0$ for distinct simple roots •
existence and characterization of bases •
Weyl chambers •
lemmas on simple roots.

#10: April 22

Reference: [1] Sec. 10

Contents
The simply transitive action of the Weyl group •
Irreducible root systems •
unique maximal root •
$W$ acts irreducibly on $E$ •
at most $2$ root lengths •
the maximal root is long.

#11: April 29

Reference: [1] Sec. 11-12

Contents
Classification •
Cartan matrix •
Coxeter graphs and Dynkin diagrams •
classification theorem •
construction of root systems.

#12: May 6

Reference: [1] Sec. 14

Contents
Reduction to the simple case •
simple Lie algebras have irred. root systems •
root system structure of semisimple Lie algebras •
isomorphism theorem •
Chevalley basis •
automorphisms.