Lecture 6 - Group Theory Flashcards
For a set and composition law to be a group, G, what must it satisfy?
Must be closed
Must have an identity
Must have an inverse
Must be associative
What does closed mean?
If g1, g2 are elements of G, then g1•g2 is also an element of G
What is the identity?
Denoted e,
e•g = g•e = g, if and only if g is an element of G
What is the inverse?
g^{-1}•g = g•g^{-1} = e
What does being associative mean?
g1•(g2•g3) = (g1•g2)•g3
Definition of a group
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
Definition of an Abelian group?
For all elements in G,
g1•g2 =g2•g1
i.e. Commutative
Definition of a Lie group?
Essentially it is a group where the elements are continuous.
Example of a Lie group?
U(1)
Elements of general linear group?
GL(n,R) is defined to consist of elements A where,
A is a real nxn matrix
det(A) != 0
Number of dimensions for GL(n,R)?
n^2, as there are n rows and n columns
Definition of special linear group?
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
Definition of orthogonal group?
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
Dimension of O(n,R)?
Bonus points for why
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.
What are the elements of SO(n,R)?
Special orthogonal group consists of elements A, where
A must be an element of O(n,R)
And det(A) = 1
Dimensions of SO(n,R)
bonus points for why
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
How do we define elements of O(p,q,R)?
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.
What are the special cases of O(p,q,R)?
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
What is the symplectic matrix?
Ω = (0 , I)
(-I, 0)
Where I is the identity matrix
What are the elements for the symplectic group?
Sp(2n,R) consists of elements A, where A must be an element of GL(n,R)
And A’ΩA = Ω
