A Lie algebra is a vector space L over some field F (typically the real or complex numbers) together with a binary operation [·, ·] : L × L -> L, called the Lie bracket, which satisfies the following properties:
- it is bilinear, i.e. [a x + b y, z] = a [x, z] + b [y, z] and [z, a x + b y] = a [z, x] + b [z, y] for all a, b in F and all x, y, z in L.
- it satisfies the Jacobi identity, i.e. [[x, y], z] + [[z, x], y] + [[y, z], x] = 0 for all x, y, z in L.
- it is antisymmetric, i.e. [x, y] = - [y, x] for all x, y in L.
Note that the Lie bracket is not a "multiplication" in the usual sense because it is not associative.
If an associative algebra L with multiplication * is given, it can be turned into a Lie algebra by defining [x, y] = x * y - y * x. This expression is called the commutator of x and y. Conversely, it can be shown that every Lie algebra can be embedded into one that arises from an associative algebra in this fashion.
Much more important examples of Lie algebras come from analysis: the vector fields on a differentiable manifold form an infinite dimensional Lie algebra in a natural way, and the left-invariant vector fields on a Lie group can be turned into a finite dimensional Lie algebra in the same fashion. Equivalently, one may think of the underlying vector space of the Lie algebra belonging to a Lie group as the tangent space at the group's identity element. The multiplication is the differential of the group commutator, (g,h) |-> ghg-1h-1.
As a concrete example, consider the Lie group GL(n,R) of all invertible n-by-n matrices with real entries. The tangent space at the identity matrix may be identified with the space M(n,R) of all real n-by-n matrices, and the Lie algebra structure coming from the Lie group coincides with the one arising from commutators of matrix multiplication.
Lie algebras can be classified to some extent, and this classification helps in understanding Lie groups, which are the truly interesting objects in geometry, analysis and physics since they capture symmetries of analytical structures. Lie algebras were originally introduced and studied by Sophus Lie and independently by Wilhelm Killing starting in the 1870s for this reason.
- We need definitions for semisimple, simple, abelian, nilpotent, solvable Lie algebras here.