Lie Groups
(pronounced Lee Group)
What is a Lie Group - A non-rigorous readable paper
A group which contains an infinite continuum of elements. Yet its structure is delineated by a finite number of elements, known as generators, from which the elements are obtained.
For example, if we are considering rotations in 3 dimensions, we can use 3 generators:
- rotation about the x-axis.
- rotation about the y-axis.
- rotation about the z-axis.
to generate all possible rotations.
Symmetry in lie group
If a given law is symmetric, or invariant, with respect to a set of actions that form a lie group, then noethers theorem tells us there is a conserved physical quantity associated with each generator of the lie group.
Rotation Groups:
| number of generators | ||
| R(2) | The group of rotations in 2 real dimensions | 1 |
| R(3) | The group of rotations in 3 real dimensions | 3 |
| R(4) | The group of rotations in 4 real dimensions | 6 |
| U(1) | The group of rotations in 1 complex dimension | 1 |
| SU(2) | The group of rotations in 2 complex dimensions S=special 2 complex dimensions = quaternion ? Almost identical properties to R(3) except repeats after 720 degrees rather than 360 degrees. |
3 |
| SU(3) | The group of rotations in 3 complex dimensions | 8 |
| SO(3) | The Orthogonal group in 3 dimensions is denoted by O(3). SO(3) is the Special Orthogonal group which is a subgroup of O(3) with determinant +1. | |
Symmetry
