Math 651
Groups
Homework 1 - Algebras and
Due 2/22/2013
1) Consider the Lie Group SU (2), the group of 2 2 complex matrices A
T
with A A = I and det(A) = 1. The underlying set is
z w
wz
|z |2 + |w|2 = 1
(1)
with the standard S3 topology. The usual basis f
Math 651
Homework 2 - Cliord Algebras
and Weight Schemes
Due 3/15/2013
Letting (V, (, ) be a real inner product space of signature (r, s), let Ir,s
T V be the ideal generated by elements of the form x y + y x + (x, y ), and
recall that the Cliord algebra
Lecture 1 - Lie Groups and the Maurer-Cartan equation
January 11, 2013
1
Lie groups
A Lie group is a dierentiable manifold along with a group structure so that the group
operations of products and inverses are dierentiable.
Let G be a Lie group. An elemen
Lecture 4 - Lie Algebra Cohomology I
January 25, 2013
Given a dierentiable manifold M n and a k -form , recall Cartans formula for the
exterior derivative
k
(1)i xi ( (x0 , . . . , xi , . . . , xk )
d (v0 , . . . , vk ) =
i=0
(1)
(1)i+j ([xi , xj ], x0 ,
Lecture 7 - Universal Enveloping Algebras and Related
Concepts, I
February 4, 2013
1
Tensor Algebras
Unlike most other topics in our class, which generally require algebraic completeness in one
way or another, our eld F is arbitrary unless otherwise state
Lecture 10 - Representation Theory II: Heuristics
February 15, 2013
1
Weights
1.1
Weight space decomposition
We switch notation from last time. We let indicate a highest weight, and to be an
arbitrary weights themselves.
If g is simple of rank n with CSA
Lecture 13 - Characters
February 25, 2013
1
Group Characters
Given any group G and a nite-dimensional representation : G GL(V ), where V is a
vector space over F , we can dene the character of the representation : G F by
(g ) = T r (g ).
(1)
Generally is
Lecture 16 - Weyls Character Formula I: The Weyl
Function and the Kostant Partition Function
March 22, 2013
References:
A. Knapp, Lie Groups Beyond an Introduction. Ch V
Fulton-Harris, Representation Theory. Ch 24, 25
R. Cahn, Semi-Simple Lie Algebras and
Lecture 19 - Cliord and Spin Representations
April 5, 2013
References:
Lawson and Michelsohn, Spin Geometry.
J. Baez, The Octonions.
1
The Lie Algebras so(n) and spin(n)
We know the Lie algebra so(n) consists of antisymmetric matrices, which can be identi
Lecture 22 - F4
April 19, 2013
1
Review of what we know about F4
We have two denitions of the Lie algebra f4 at this point. The old denition is that it is
the exceptional Lie algebra with Dynkin diagram
1
2
3
4
Adj.
The Weyl dimension formula gives the fo
Lecture 23 - The Magic Square
April 22, 2013
References:
The Octonions. J Baez (2001)
Spin(8), Triality, F4 and all that, F. Adams (1981)
Trialities and the Exceptional Lie Algebras: Deconstructing the Magic Square. J. Evans
(2009)
Geometries, the princip
Math 651 Homework 4 - Cliord and Jordan
Due 5/7/2013
1) Prove the period 2 periodicity for the complex Cliord algebras: Cln+2
Cln C(2).
2) (Reference: Lie Groups Beyond and Introduction, A. Knapp, Ch V) This
exercise gives an explicit construction of the
Lecture 3 - Cambpell-Baker-Hausdor
January 18, 2013
Reference for this lecture: Lie Algebras, 2004, by Shlomo Sternberg
1
Statement
Set
z
log(z )
z1
(z ) =
(1)
and note that this is continuous at z = 1. In fact we have the taylor series
=
1+z
1+z
log(1 +
Lecture 6 - Lie Algebra Cohomology III
February 1, 2013
1
Application: Symplectic geometry
We have a symplectic manifold (M, ), where is a closed 2-form. Assume G is a connected
Lie group that acts on M via symplectomorphisms (meaning it preserves ), whic
Lecture 9 - Representation Theory I: Examples
February 11, 2013
1
Finite dimensional sl(2, C) modules: review
A consequence of the semisimplicity of sl(2, C) is that any element x sl(2, C) can be
written x = xs + xn , where xs , xn have the property that
Lecture 11 - Representation Theory IV: Existence
February 22, 2012
In this lecture we nally prove that representations exist, and that an irreducible
representations is nite dimensional if and only if it has a highest weight that is dominant
integral.
1
V
Lecture 15 - A Few Odds and Ends
March 18, 2012
1
1.1
Completion of the Harish-Chandra Theorem
The Setup
A semisimple Lie algebra g of rank n with choice of CSA h, leading to a Borel subalgebra
b and corresponding positive an negative nilpotent algebras n
Lecture 18 - Cliord Algebras and Spin groups
April 5, 2013
Reference: Lawson and Michelsohn, Spin Geometry.
1
Universal Property
If V is a vector space over R or C, let q be any quadratic form, meaning a map q : V W
with q (v ) = |2 q (v ) for scalars . G
Math 651
Projections
Homework 4 - Centers and
Due 4/12/2013
Recall the decomposition
U (g) = U (h) U (g)n+ + n U (g)
(1)
and the associated projection : U (g) U (h). Recall the twist map
: U (h) U (h) given on generators by (h) = h (h)1, and the HarishCh
Lecture 2 - Lie Groups, Lie Algebras, and Geometry
January 14, 2013
1
Overview
If D is any linear operator on a vector space, we can dene Exp(D) by
Exp(D) =
1n
D.
n!
n=0
(1)
The sum converges if the operator is bounded. In other cases, such as dierential
Lecture 5 - Lie Algebra Cohomology II
January 28, 2013
1
Motivation: Left-invariant modules over a group
Given a vector bundle F G over G where G has a representation on F , a left Gaction on is a map that commutes with left-multiplication on G itself: fo
Lecture 8 - Universal Enveloping Algebras and Related
Concepts, II
February 8, 2013
1
Filtrations and Graded Algebras
1.1
The basics
An algebra U has a ltration if there are subsets cfw_U (i) iZ of U so that
i) U (i) U (i+1) . . .
ii) U(i) U (j ) U (i+j )
Lecture 10 - Representation Theory III: Theory of
Weights
February 18, 2012
1
Terminology
One assumes a base
= cfw_i i has been chosen. Then a weight with non-negative
integral Dynkin coecients i = , i is called a dominant weight. If all coecients are
all
Lecture 14 - Isomorphism Theorem of Harish-Chandra
March 11, 2013
This lectures shall be focused on central characters and what they can tell us about
the universal enveloping algebra of a semisimple Lie algebra.
1
Invariant Polynomials
In our discussion
Lecture 17 - Weyls Character Formula II: Formulas of
Weyl and Kostant
March 22, 2013
1
The Weyl Character Formula
We now get down to the business of proving the Weyl formula. The idea is that, having
a formula for the alternating function Q, we develop a
Lecture 20 - Duality and Triality
April 8, 2013
References:
Spin Geometry. M. Michelsohn and B. Lawson (1989)
The Octonions. J. Baez (2001)
1
Review
1.1
Representations
Letting Rn have a positive denite inner product and using the notation
for the exterio
Lecture 21 - Jordan Algebras and Projective Spaces
April 15, 2013
References:
Jordan Operator Algebras. H. Hanche-Olsen and E. Stormer
The Octonions. J. Baez
1
Jordan Algebras
1.1
Denition and examples
In the 1930s physicists, looking for a larger context
Special Lecture - The Octionions
March 15, 2013
1
1.1
R
Denition
Not much needs to be said here. From the God given natural numbers, we algebraically
build Z and Q. Then create a topology from the distance function (subtraction) and complete Q to obtain R