A Lie algebra is simply
- a) a vector space
- b) with an “operation” such that the operation [x,y] of any two vectors x and y is again a vector, and such that the following hold:
- skew-symmetry: [x,y] = -[y,x].
- Bi-linearity: [x,ay] = a[x,y], [x,y+z] = [x,y] + [x,z]. (a is a number.)
- Jacobi identity: [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0.
The best known example is the Vector Space R3 with the cross product as the operation.
The real importance of Lie algebras is that one can get one from any Lie group – roughly speaking, a group that’s also a manifold
Lie groups crop up as the PRIMARY groups of SYMMETRIES in physics. The Lie algebra is essentially the infinitesimal version of the corresponding Lie group.
E.g. – The relation between the group of rotations in R3 and the cross product.
Here the group is called SO(3) and the Lie algebra is called so(3).
(So R^3 with its cross product is called so(3).) One can generalize this to any number of dimensions, letting SO(n) denote the group of rotations in R^n and so(n) the corresponding Lie algebra. (However, so(n) is not isomorphic to R^n except for n = 3, so there is something very special about three dimensions.)