Lie Groups
(pronounced Lee Group)
A combination of infinite groups and calculus. For example 'infinitesimal elements' allow us to build rotations by integrating some infinitesimal rotation. A group that has infinitesimal generators is called a continuous group.
Lie groups turn up when we study a geometric objects with a lot of symmetry, such as a sphere, a circle, or flat spacetime. Because there is so much symmetry, there are many functions from the object to itself that preserve the geometry and these functions become the elements of the group. Discrete groups can also be used to keep track of symmetries.
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. Such as rotation group where we can build rotations by integrating infinitesimal rotations. Lie groups are therefore a combination of calculus and group theory. Functions from the group preserve symmetry and these functions become the elements of the group.
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
