Lie groups and Lie algebras

Lie groups can be illustrated by continuous rigid motion of a ball with painted patterns on it in 3-dimensional space. The Lie group ISO(3) consists of all rigid transformations.

A rigid transformation is essentially the act of picking the ball and placing it somewhere else, ignoring the detailed motion in between and the location one started. Special transformations are for example a translation in northern direction by 1 meter, or a rotation by one quarter around the vertical axis at some particular point (think of a ball with a string attached). 'Rigid' means that the distances between marked points on the ball remains the same; the mathematician talks about 'preserving distances', and the distances are therefore labeled 'invariants'.

One can repeat the same transformation several times, or two different transformations and get another one - This is called the product of these transformations. For example, the product of a translations by 1 meter and another one by 2 meters in the same direction gives one of 1+2=3 meters in the same direction. In this case, the distances add, but if one combines rotations about different axes the result is no longer intuitive. To make this more tractable for calculations, one needs to take some kind of logarithms of transformations - these behave again additively and make up the corresponding Lie algebra iso(3) [same letters but in lower case]. The elements of the Lie algebra can be visualized as very small, or 'infinitesimal', motions.

General Lie groups and Lie algebras extend these notions to to more general manifolds. A manifold is just a higher-dimensional version of space, and transformations are generalized motions preserving invariants that are important in the manifold. The transformations preserving these invariants are also called 'symmetries', and the Lie group consisting of all symmetries is called a 'symmetry group'. The elements of the corresponding Lie algebra are 'infinitesimal symmetries'.

For example, physical laws are invariant under rotations and translations, and hence unter all rigid motions. But not only these: If one includes time explicitly, the resulting 4-dimensional space has more invariant motions or ''symmetries''. The Lie group of all these symmetry transformations is called the Poincar'e group, and plays a basic role in the theory of relativity. The transformations are now about space-time frames in uniform motion. Apart from translations and rotations there are symmetries called 'boosts' that accelerate a frame in a certain direction, and combinations obtained by taking products. All infinitesimal symmetries together make up a Lie algebra, called the Poincar'e algebra.

Much more on Lie groups and Lie algebras from the perspective of classical and quantum physics can be found in:

Arnold Neumaier and Dennis Westra,
Classical and Quantum Mechanics via Lie algebras,
arXiv:0810.1019