Lecture 21 - Cartan and Engel Subalgebras (Book: ch 15)
November 27, 2012
1
Inner Automorphisms
2
Engel Subalgebras
Arbitrary (nite dimensional) algebra L over C.
Let x L. Generalized eigenspace decomposition: La (ad x) characterized by the maximality
m
o
Lecture 17 - The Weyl Group
November 1, 2012
1
Information on Weyl groups
Theorem 1.1 Let
be a base of
a) If E is regular, there exists some W so that ( ( ), ) > 0 for all
particular W acts transitively on Weyl chambers
b) If
is any other base, there is
Lecture 16 - Simple Roots
October 30, 2012
1
Existence of a base
Recall that a subset
B1)
is called a base if
is a basis of E
B2) Each root can be written as a sum =
c where
k are either positive or all negative (or zero).
If
is a base for then we call
Lecture 15 - Root System Axiomatics
Nov 1, 2012
In this lecture we examine root systems from an axiomatic point of view.
1
Reections
If v Rn , then it determines a hyperplane, denoted Pv , through the origin. Reection
about this hyperplane, denoted v , is
Lecture 14 - o(4) and g2
October 23, 2012
In this lecture we take a closer look at the orthogonal algebras.
1
Example: o(4)
1.1
Identication with an alternating algebra
Given a Riemannian metric g (, ) on any vector space V , there is are two a bilinear m
Lecture 13 - Root Space Decomposition II
October 18, 2012
1
Review
First let us recall the situation. Let g be a simple algebra, with maximal toral subalgebra
h (which we are calling a CSA, or Cartan Subalgebra). We have that h acts on g via the
adjoint a
Lecture 12 - Root Space Decomposition
October 16, 2012
1
Review
First let us recall the situation. Let g be a simple algebra, with maximal toral subalgebra
h (which we are calling a CSA, or Cartan Subalgebra). We have that h acts on g via the
adjoint acti
Lecture 11 - Cartan Subalgebras
October 11, 2012
1
Maximal Toral Subalgebras
A toral subalgebra of a Lie algebra g is any subalgebra consisting entirely of abstractly
semisimple elements.
Lemma 1.1 If t g is any toral subalgebra, then t is abelian.
Pf. As
Lecture 10 - Examples of sl(2, C)-modules
Oct 9, 2012
Let H = L2 (R3 ) be the space of complex-valued L2 -functions on R3 , and let S = L2 (S2 )
be the space of complex-valued L2 -functions on S2 .
1
Preliminaries: Riemannian geometry of S2
To coordinatiz
Lecture 9 - Structure of sl(2, C)-modules
Oct 4, 2012
1
Structure of sl(2, C)-modules
We use the standard basis sl2 = spancfw_x, y, h with
[x, y ] = h [h, x] = 2x [h, y ] = 2y.
(1)
Note that x and y are abstractly nilpotent and h is abstractly semisimple.
Lecture 8 - Preservation of the Jordan Decomposition
and Levis Theorem
Oct 2, 2012
1
Preservation of the Jordan decomposition
Theorem 1.1 Assume g gl(V ) is a semisimple linear Lie algebra. Given any x g, the
abstract and usual Jordan decompositions coinc
Lecture 7 - Complete Reducibility of Representations of
Semisimple Algebras
September 27, 2012
1
New modules from old
A few preliminaries are necessary before jumping into the representation theory of semisimple algebras. First a word on creating new g-mo
Lecture 6 - Structure of Semisimple Lie Algebras
September 25, 2012
1
The abstract Jordan decomposition for semisimple
Lie algebras
Recall that a derivation of a Lie algebra g is a map End(V ) so that [x, y ] = [x, y ] +
[x, y ] for all x, y g.
Propositio
Lecture 5 - Cartans criterion and semisimplicity
September 20, 2012
1
Cartans Criterion
Theorem 1.1 Assume the base eld is C, and let A B be vector subspaces of End(V ).
Set M = cfw_x End(V ) | [x, B ] A. If x M and T r(xy ) = 0 for any other y M , then
x
Lecture 4 - The Fitting and Jordan-Chevalley
decompositions
September 18, 2012
1
Minimal and Characteristic polynomials
If x End(V ), its minimal polynomial M is the monic polynomial of smallest degree so
that the transformation M (x) End(V ) is zero. The
Lecture 3 - Lies Theorem
September 13, 2012
1
Weights and Weight Spaces
Proposition 1.1 If L is a vector space of endomorphisms of V and v V is an eigenvector
common to all endomorphisms x L, then v determines a linear operator v : L F,
dened implicitly b
Lecture 2 - Fundamental denitions, and Engels
Theorem
September 11, 2012
1
Basic Denitions
A representation of a Lie algebra L is a homomorphism of L into the Lie algebra gl(V ) for
some vector space V over F. Every Lie algebra has at least one representa
Lecture 1 - Basic Denitions and Examples of Lie
Algebras
September 6, 2012
1
Denition
A Lie algebra l is a vector space V over a base eld F, along with an operation [, ] : V V
V called the bracket or commutator that satises the following conditions:
Bil
Math 650
Homework 3
Due Dec 19
1) 14 #5
2) 15 #1
3) Let g = sp(4, C) and h a toral subalgebra of your choice, spanned by
some matrices t1 , t2 sp(4, C). Show that a complimentary subspace is
spanned by (abstractly) nilpotent elements. Determine ad h : g g
Math 650
Homework 2
Due 10/25/2012
1) 4, # 1
2) 4, # 4
3) 4, # 5
4) 5 # 2
5) Let h sl(3, C) be the subalgebra of diagonal matrices, spanned by your
choice of h1 , h2 sl(3.C). Show that h consists of (abstractly) semisimple
elements, that h is abelian, and