Lecture 6 - Group Theory

Question 1

For a set and composition law to be a group, G, what must it satisfy?

Answer 1

Must be closed
Must have an identity
Must have an inverse
Must be associative

Question 2

What does closed mean?

Answer 2

If g1, g2 are elements of G, then g1•g2 is also an element of G

Question 3

What is the identity?

Answer 3

Denoted e,

e•g = g•e = g, if and only if g is an element of G

Question 4

What is the inverse?

Answer 4

g^{-1}•g = g•g^{-1} = e

Question 5

What does being associative mean?

Answer 5

g1•(g2•g3) = (g1•g2)•g3

Question 6

Definition of a group

Answer 6

A set of objects, supplied with a rule that allows you to take two objects and make another one.

This law of composition is denoted with a dot:
g1 •g2 = g3

Question 7

Definition of an Abelian group?

Answer 7

For all elements in G,
g1•g2 =g2•g1

i.e. Commutative

Question 8

Definition of a Lie group?

Answer 8

Essentially it is a group where the elements are continuous.

Question 9

Example of a Lie group?

Answer 9

U(1)

Question 10

Elements of general linear group?

Answer 10

GL(n,R) is defined to consist of elements A where,

A is a real nxn matrix
det(A) != 0

Question 11

Number of dimensions for GL(n,R)?

Answer 11

n^2, as there are n rows and n columns

Question 12

Definition of special linear group?

Answer 12

SL(n,R) consists of elements A where,
A is an element of GL(n,R)
det(A) = 1

It has n^2 -1 elements, as there are n rows and n columns and one real constraint

Question 13

Definition of orthogonal group?

Answer 13

O(n,R) consists of elements A where,
A is an element of GL(n,R)
AA’ = A’A = I, where ‘ represents the transpose and I is the identity matrix

Question 14

Dimension of O(n,R)?

Bonus points for why

Answer 14

1/2 * n(n-1)

As every diagonal element needs to be 1, there are n constraints. Also every off-diagonal element needs to be 0, which is n^2 -n constraints but as this is symmetric there are only 1/2(n^2 - n) constraints from this.

Question 15

What are the elements of SO(n,R)?

Answer 15

Special orthogonal group consists of elements A, where
A must be an element of O(n,R)
And det(A) = 1

Question 16

Dimensions of SO(n,R)

bonus points for why

Answer 16

The same as O(n,R), 1/2 n(n-1)

Imposing the unit determinant only selects a sector of O(N,R) rather than reducing its dimension as we already have det(A)= +- 1 for orthogonal matrices

Question 17

How do we define elements of O(p,q,R)?

Answer 17

Define elements of O(p,q,R) by requiring:
AηA’ = η
where ‘ means the transpose, and η = diag( -1, -1, …, 1, 1 …), with p minus ones and q plus ones.

Question 18

What are the special cases of O(p,q,R)?

Answer 18

p=0, for any q, η represents the identity matrix, and therefore O(0,q,R) is equal to O(n,R).

p=1, q=3, is the Lorentz group for special relativity

Question 19

What is the symplectic matrix?

Answer 19

Ω = (0 , I)
(-I, 0)
Where I is the identity matrix

Question 20

What are the elements for the symplectic group?

Answer 20

Sp(2n,R) consists of elements A, where A must be an element of GL(n,R)
And A’ΩA = Ω