A Lie group G is an analytic manifold that is also a group such that the group operations multiplication and inversion are analytic. Lie groups are important in analysis, physics and geometry because they serve to describe the symmetry of analytical structures. Lie groups (pronounced as "lee") were introduced by Sophus Lie in 1870 in order to study symmetries of differential equations.
A vector field X on a Lie group G is said to be left invariant if it commutes with left translation: Define Lg[f](x)= f(gx) for any analytic function f : G -> R and all g, x in G. Then a vector field is left invariant if X Lg = Lg X for all g in G.
The set of all vector fields on an analytic manifold is a Lie algebra. The subalgebra of all left invariant vector fields is called the Lie algebra associated with G, and is usually denoted by a gothic g. This Lie algebra g is finite-dimensional (it has the same dimension as the manifold G) which makes it susceptible to classification attempts. By classifying g, one can also get a handle on the Lie group G.
Every element v of the tangent space Te at the identity element e of G determines a unique left invariant vector field whose value at the element x of G will be denoted by xv; the vector space underlying g may therefore be identified with Te. The vector v furthermore determines a function c : R -> G whose derivative everywhere is given by the corresponding left invariant vector field
- c'(t) = c(t) v
and which has the property
- c(s + t) = c(s) c(t)
for all s and t. The operation on the right hand side is the group multiplication in G. The formal similarity of this formula with the one valid for exponential functions justifies the definition
- exp(v) = c(1)
This is called the exponential map and it maps the Lie algebra g into the Lie Group G. It provides a diffeomorphism between a neighborhood of 0 in g and a neighborhood of e in G.
The operation t, s -> tst-1s-1 is called the commutator operator. Note that this operation sends (1,1) to 1, and so defines an operation on the tangent space at 1 by differentiation. The function it defines on the tangent space is bi-linear, and it turns out that it gives the tangent space a Lie algebra structure.
Sometimes, Lie groups are defined as topological manifolds with continuous group operations; the two definitions are equivalent. This is the content of Hilbert's fifth problem. The precise statement, proven by Gleason, Montgomery and Zippin in the 1950s, is as follows: If G is a topological manifold with continuous group operations, then there exists exactly one differentiable structure on G which turns it into a Lie group in our sense.