Index Notation
J.Pearson
July 18, 2008
Abstract
This is my own work, being a collection of methods & uses I have picked up. In this work, I
gently introduce index notation, moving through using the summation convention. Then begin
to use the notation, thr
LIE ALGEBRAS
KEVIN MCGERTY 2012, WITH MINOR MODIFICATIONS BY BALAZS SZENDROI 2013
1. B ACKGROUND
In this section I use some material, like multivariable analysis, which is not necessary for the main body of the
course, but if you know it (and hopefully if
C2.1a Lie algebras
Mathematical Institute, University of Oxford
Michaelmas Term 2014
Problem Sheet 6
Throughout this sheet we assume that all Lie algebras and all representations discussed are nite dimensional unless the contrary is explicitly stated, and
C2.1a Lie algebras
Mathematical Institute, University of Oxford
Michaelmas Term 2014
Problem Sheet 5
Throughout this sheet we assume that k is an algebraically closed eld of characteristic zero, and all Lie
algebras and representations discussed are nite
C2.1a Lie algebras
Mathematical Institute, University of Oxford
Michaelmas Term 2014
Problem Sheet 2
1. Find the structure constants of sl2 with respect to the basis
h=
1 0
0 1
,
0
0
e=
1
0
f=
,
0
1
0
0
.
Show that the only ideals of sl2 (C) are 0 and its
C2.1a Lie algebras
Mathematical Institute, University of Oxford
Michaelmas Term 2014
Problem Sheet 4
Assume throughout the problems that we work over a eld k which is algebraically closed of characteristic zero, unless the contrary is explicitly stated.
1
C2.1a Lie algebras
Mathematical Institute, University of Oxford
Michaelmas Term 2014
Problem Sheet 3
All Lie algebras and all representations are nite dimensional unless otherwise stated.
1. Let k be an innite eld (not necessarily algebraically closed or
C2.1a Lie algebras
Mathematical Institute, University of Oxford
Michaelmas Term 2014
Problem Sheet 1
Throughout this sheet we assume that all Lie algebras are over a eld k.
There is an algebraic was to think about the idea of innitesimals. The rst two que
Chapter 1
Preliminaries
These are lecture notes, not a textbook. They are not meant to replace going to
the lectures. Specically, they do not contain any motivation for the concepts
nor an intuitive explanation. They also may be out of order with the lect
Problem Sheet 3
March 17, 2014
1. Connectedness
(a) If f : X Y is continuous and Y is disconnected as witnessed by
a continuous surjection g : Y 2 then g f witnesses that X is
disconnected.
(b) : apply the previous with the projections i .
: It is enough
Problem Sheet 3
Max Pitz, Rolf Suabedissen
March 17, 2014
1. Convergence and Filters
(a)
i. x A = A#Nx = A#Nx x and A#N can be extended
to some ultralter U which is as required. Conversely, if U is an
ultralter containing A and converging to x then U Nx a
Problem Sheet 3
November 25, 2013
1. Connectedness
(a) Note that continuous images of connected topological spaces are connected.
(b) Show that a nite product of topological spaces is connected if and
only if every factor is connected.
(c) Suppose Xi , i
1
Linearly Ordered Spaces
1. Separation Axioms in a LOTS First note that cfw_(x, y) : x < y is a
basis for the order topology and that for x < y (, x) (y, ) = if
and only if (x, y) = , i.e. there is no r with x < r < y. Also note that
[x, y] = X \ (, x) (
Problem Sheet 3
Max Pitz, Rolf Suabedissen
November 15, 2013
1. Convergence and Filters
(a) Check all the unproven statements about lters and convergence from
the lecture notes. In particular:
i. x A i there is a lter F A converging to x i there is an
ult
1
Linearly Ordered Spaces
A linear (or total) order on a set is a transitive, reexive relation such that
x y and y x implies x = y.
If is a linear order on a set X and x X, we write (, x) = cfw_y X : y x, y = x
and (x, ) = cfw_y X : x y, x = y. We may use
Solutions to Sheet 1
Rolf Suabedissen
October 25, 2013
Warning: These are not model solutions. They might even be incorrect
(sometimes badly so - I hope they are not). If you nd the inevitable mistakes,
let me know and I might x them.
1. Countable and Unc
DIFFERENTIABLE MANIFOLDS
C3.3 Course 2015
Solutions to Question Sheet 7
1.
F (1 1 + 2 2 ) = 1 F 1 + 2 F 2
and integration of forms is linear, so
N
F
is a linear map on all forms. Now if = d then
N
F (d) =
N
dF () = 0
by Stokes theorem, so the linear map
DIFFERENTIABLE MANIFOLDS
C3.3 Course 2015
Question sheet 2
1. Show that the product M N of two manifolds is a manifold.
2. Let A : Rn+1 Rn+1 be an invertible linear transformation. Show that A maps the set
RPn of 1-dimensional subspaces of Rn+1 bijectivel
DIFFERENTIABLE MANIFOLDS
C3.3 Course 2015
hitchin@maths.ox.ac.uk
Question sheet 4
1. Take p V where dim V = n and consider the linear map A : np V n V dened
by A () = .
(i) Show that if = 0, then A = 0.
(ii) Prove that the map A is an isomorphism from p V
Tensor Calculus 2 - Differentiation
Now, the equation of a geodesic is:
&
x i + i jk x j x k = 0
& &
Where:
dx i
&
xi =
etc
ds
Now, transform:
x i = x i ( x )
s = s
x i x r
x i r
&
&
xi =
=
x
x r s
x r
d x i
&
xi = r xr
x &
ds
Notice that:
d x i
x i
Tensor Calculus 1 Tensors & Geodesics
Tensor Calculus:
Riemannian Space:
Suppose you have two points in normal Euclidean space:
P( y1 , y 2 , y 3 )
Q( y1 + dy1 , y 2 + dy 2 , y3 + dy 3 )
Now, the distance between the two points is given by:
2
2
ds 2 = dy1
1
Derivation of Lagrange Equations
Consider a particle acted upon by forces X, Y, Z. By Newtons 2nd law:
m = X
x
(1)
m = Y
y
(2)
m = Z
z
(3)
x = x (q1 , q2 , q3 , . . . , qn , t)
(4)
y = y (q1 , q2 , q3 , . . . , qn , t)
(5)
z = z (q1 , q2 , q3 , . . . ,
Lagrangian Dynamics 2 - Calculus of Variations & Hamiltons Principle
The work done when q i changes to q i + qi for i = 1,2,., n is given by:
n
W = Qiqi
(1)
i =1
We can find Qi from (1) when the system is not conservative.
Calculus of Variations:
Consider
Lagrangian Dynamics 3 Hamiltons Equations, Differentials & Small Oscillations
Hamiltons Equations:
d L L
=0
&
dt qi qi
Putting:
L
= pi
&
q i
dpi L
=0
dt q i
L
&
pi =
qi
Now:
L
&
&
pi =
=
L (q , q, t ) = f i ( q, q , t )
&
&
qi qi
&
There are n pi s and
ELECTROMAGNETISM PHYS20141 2014/15 : Prof R.A. Battye
EM WAVES IN A NUTSHELL
Only derivations denoted with a (*) will be tested in the examination.
This is completely new material.
In this part of the the course we will use x, y and z to denote unit vec
ELECTROMAGNETISM PHYS20141 2014/15 : Prof R.A. Battye
MAXWELLS EQUATIONS IN A NUTSHELL
Only derivations denoted with a (*) will be tested in the examination. Typically you will
not be asked to remember complicated mathematical results, but you will be exp
ELECTROMAGNETISM PHYS20141 2014/15 : Prof R.A. Battye
MATERIALS IN A NUTSHELL
Only derivations denoted with a (*) will be tested in the examination.
This is completely new material.
Drift velocity, Ohms law and Conductors
Ohms Law:
j = E
NB. E = 0 in a pe
ELECTROMAGNETISM PHYS20141 2014/15 : Prof R.A. Battye
MATHS FOR EM IN A NUTSHELL
Only derivations denoted with a (*) will be tested in the examination. Typically you will
not be asked to remember complicated mathematical results, but you will be expected
DIFFERENTIABLE MANIFOLDS
C3.3 Course 2015
hitchin@maths.ox.ac.uk
Question sheet 6
1. Let F : S 3 S 2 be a smooth map and 2 (S 2 ) a form representing a non-trivial de
Rham class w H 2 (S 2 ). Show that we can nd a 1-form on S 3 such that F () = d.
Show fu