Lie Groups and Algebras

Now it is time for some mathematical background.

A group is a set of objects that can be multiplied and have an inverse. 

A Lie group is a group that depends on continuous parameters. The name comes from Norwegian mathematician Sophus Lie and is pronounced Lee.

The simplest Lie group is the special orthogonal group SO(2), which describes rotations in the 2D plane. It is isomorphic (essentially the same as) the group U(1) of 1x1 unitary matrices, which can be identified with complex numbers of unit length. A rotation by an angle \(\varphi\) in the plane can be identified with multiplication by the complex number \(\exp(i\varphi)\).

A Lie group that is umbiquous in 3D software is the special orthogonal group SO(3), which describes rotations in 3D space. This is the group of real 3x3 matrices with unit determinant, where the inverse is the transposed matrix.

The special unitary group SU(2) of complex 2x2 matrices is the double cover of SO(3). This means that the groups are locally the same, but two separate SU(2) matrices correspond to the same SO(3) matrix. Quaternions are SU(2) matrices, and hence there are two different quaternions that correspond to a given rotation in 3D space.

A Lie algebra is the infinitesimal form of a Lie group. The basic operation in a Lie algebra is the commutator (or Lie bracket), given by
\[
[A, B] = AB - BA
\]
From the definition and associativity of matrix multiplication (\((AB)C = A(BC) = ABC\)) follow two conditions that any Lie algebra must obey.

Anti-symmetry: 
\[
[B, A] = -[A, B].
\]
Jacobi identify:
\[
[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0.
\]

The infinitesimal form of the group SU(2) is the Lie algebra su(2), with three elements \(\sigma_i, i = 1,2,3\) (the Pauli matrices) that satisfy the brackets
\[
[\sigma_i, \sigma_j] = \sum_{k=1}^3 \epsilon_{ijk}\ \sigma_k,
\]
where \(\epsilon_{ijk}\) equals \(1\) if \(ijk\) is a positive permutation of \(123\), \(-1\) if it is a negative permutation, and \(0\) if two indices are equal. It is straightforward to verify that this bracket obeys anti-symmetry and the Jacobi identities.

A group which will play a prominent role in the future is the diffeomorphism group, i.e. the group of invertible maps of \(d\)-dimensional space to itself. Let \(F\) and \(G\) be two such maps. They form a group, with multiplication given by composition
\[
(F \circ G)(x) = F(G(x))
\]
The associated Lie algebra is the algebra of vector fields, which I informally call the diffeomorphism algebra. A vector field is locally of the form \(\xi = \xi^\mu(x) \partial_\mu\), where \(\partial_\mu = \partial/\partial x^\mu\) is the partial derivative, and I use Einstein's summation convention: repeated indices, one up and one down, are implicitly summed over.

The diffeomorphism algebra is defined by the brackets
\[
[\xi, \eta] = \xi^\mu \partial_\mu \eta^\nu \partial_\nu 
- \eta^\nu \partial_\nu \xi^\mu \partial_\mu.
\]
It is straightforward to verify that this is a Lie algebra (anti-symmetry and Jacobi identifies).

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