Fourier Analysis on SU(2)

125 Pages·2012·0.477 MB·English

by  Tyler Leaser

ABSTRACT
FOURIER ANALYSIS ON SU(2)
by
Tyler Leaser
November, 2012
Chair: Dr. Chal Benson
Major Department: Mathematics
The set SU(2) of 2x2 unitary matrices with determinant one forms a compact non-
abelian Lie group diffeomorphic to the three dimensional sphere. This thesis surveys
general theory concerning analysis on compact Lie groups and applies this in the
setting of SU(2). Our principal reference is J. Faraut’s book Analysis on Lie Groups.
Fundamental results in representation theory with compact Lie groups include the
Peter-Weyl Theorem, Plancherel Theorem and a criterion for uniform convergence of
Fourier series.
On SU(2) we give explicit constructions for Haar measure and all irreducible
unitary representations. For purposes of motivation and comparison we also consider
analysis on U(1), the unit circle in the complex plane. In this context, the general
theory specializes to yield classical results on Fourier series with periodic functions
and the heat equation in one dimension. We discuss convergence behavior of Fourier
series on SU(2) and show that Cauchy problem for the heat equation with continuous
boundary data admits a unique solution.

FOURIER ANALYSIS ON SU(2)
A Thesis
Presented to
The Faculty of the Department of Mathematics
East Carolina University
In Partial Fulfillment
of the Requirements for the Degree
Master of Arts in Mathematics
by
Tyler Leaser
November, 2012
Copyright 2012, Tyler Leaser
FOURIER ANALYSIS ON SU(2)
by
Tyler Leaser
APPROVED BY:
DIRECTOR OF THESIS:
Dr. Chal Benson
COMMITTEE MEMBER:
Dr. Chris Jantzen
COMMITTEE MEMBER:
Dr. Imre Patyi
COMMITTEE MEMBER:
Dr. Gail Ratcliff
CHAIR OF THE DEPARTMENT
OF MATHEMATICS:
Dr. Johannes Hattingh
DEAN OF THE
GRADUATE SCHOOL:
Dr. Paul Gemperline
ACKNOWLEDGEMENTS
Apart from the efforts of myself, the success of any project depends largely on the
encouragement and guidelines of many others. I take this opportunity to express my
gratitude to the people who have been instrumental in the successful completion of
this project. I would like to show my greatest appreciation to Prof. Chal Benson. I
can’t say thank you enough for his tremendous support and help. I feel motivated and
encouragedeverytimeIattendhismeeting. Withouthisencouragementandguidance
this project would not have materialized. The guidance and support received from
all the members who contributed and who are contributing to this project, was vital
for the success of the project. I am grateful for their constant support and help.
TABLE OF CONTENTS
1 Introduction to Matrix Groups . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Matrix Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Adjoint Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Haar Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Compact Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1 Schur Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 The Peter-Weyl Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 The Plancherel Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Absolute and Uniform Convergence of Fourier Series . . . . . . . . . . 24
2.5 Fourier Series for Central Functions . . . . . . . . . . . . . . . . . . . 27
2.6 Laplace Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.7 Uniform Continuity on Compact Groups . . . . . . . . . . . . . . . . 38
3 Analysis on U(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1 Determination of U(cid:100)(1) . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Classical Fourier Series: L2-theory . . . . . . . . . . . . . . . . . . . . 43
3.3 Classical Fourier Series: Point-wise Convergence . . . . . . . . . . . . 44
3.4 Abel Summability on U(1) . . . . . . . . . . . . . . . . . . . . . . . . 45
4 Analysis on SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1 Haar Measure on SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Central Functions on SU(2) . . . . . . . . . . . . . . . . . . . . . . . 57
4.3 Irreducible Representations of SU(2) . . . . . . . . . . . . . . . . . . 60
4.4 An Alternate Model for π . . . . . . . . . . . . . . . . . . . . . . . . 78
m
4.5 Reproducing Kernels for P and V . . . . . . . . . . . . . . . . . . 82
m m
4.6 Laplace Operator on SU(2) . . . . . . . . . . . . . . . . . . . . . . . 84
4.7 Fourier Series on SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.8 Poisson Kernel on SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.9 Abel Summability of Fourier Series on SU(2) . . . . . . . . . . . . . . 98
5 Heat Equation on U(1) and SU(2) . . . . . . . . . . . . . . . . . . . . . 102
5.1 Heat Equation on U(1) . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.2 Heat Equation on SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . 104
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
CHAPTER 1: Introduction to Matrix Groups
1.1 Matrix Groups
Let M(n,K) be the set of n × n matrices with entries in K = R or K = C. Note
that M(n,K) is a vector space over K in the usual way, isomorphic to Kn2. We equip
M(n,K) with its usual vector topology. The subset of invertible matrices in M(n,K)
formsagroupundermatrixmultiplication,knownasthegenerallineargroup,denoted
GL(n,K). Another way to define the general linear group is
GL(n,K) = {X ∈ M(n,K) : det(X) (cid:54)= 0}.
As GL(n,K) = det−1(K−{0}) and det : M(n,K) → K is continuous, GL(n,K) is an
open subset of M(n,K).
Definition 1.1. A closed subgroup of GL(n,K) is called a linear Lie group.
There are a number of classic examples.
• SL(n,K) = {g ∈ GL(n,K) : det(g) = 1} = Ker(det : GL(n,K) → K) is
called the special linear group.
• O(n) = {g ∈ GL(n,R) : (cid:107)gx(cid:107) = (cid:107)x(cid:107), for all x ∈ Rn} is the orthogonal
group and SO(n) = {g ∈ O(n) : det(g) = 1} = O(n)∩SL(n,R) the special
orthogonal group. As is well known g ∈ O(n) if and only if g(cid:124)g = I.
• U(n) = {g ∈ GL(n,C) : g∗g = I} is the unitary group and
SU(n) = {g ∈ U(n) : det(g) = 1} the special unitary group.
An important fact about linear Lie groups is that each such group G ⊂ GL(n,K)
is a smooth sub-manifold of M(n,K) [2, Corollary 3.3.5]. This will allow us to apply
calculus on manifolds to linear Lie groups.
We can extend the usual exponential map to a matrix valued function of a matrix
variable.
2
Definition 1.2. The exponential of a matrix X ∈ M(n,K) is the sum of the series
(cid:88)∞ Xk
exp(X) = .
k!
k=0
Here we have that Xk is simply X matrix multiplied by itself k times with the
usual convention that X0 = I. Since matrices do not usually commute, we do not
have, in general, that exp(X +Y) = exp(X)exp(Y). However
exp(X)exp(Y) = exp(X +Y) provided XY = YX; (1.1)
see [2, Proposition 2.2.3]. Indeed, if X,Y commute we can write
∞ ∞
(cid:88)(cid:88) 1
exp(X)exp(Y) = XkY(cid:96)
k!(cid:96)!
k=0 (cid:96)=0
(cid:32) (cid:33)
∞ ∞
(cid:88) 1 (cid:88) m!
= XkY(cid:96)
m! k!(cid:96)!
(cid:96)=0 k+(cid:96)=m
∞
(cid:88) 1
= (X +Y)m
m!
m=0
= exp(X +Y).
We will also need the following important relation later.
Lemma 1.3. For X ∈ M(n,K) we have
det(exp(X)) = etr(X).