Lie group
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.
While the Euclidean space Rn is a Lie group, more typical examples are groups of invertible matrices, for instance the group SO(3) of all rotations in 3-dimensional space.
The Lie algebra associated to a Lie group
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 over the real numbers. 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. The representation theory of simple Lie Groups is the best and most important example.
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 Lie algebra structure on Te can also be described as follows : the commutator operation
- (t, s) -> tst-1s-1
sends (e,e) to e, so its derivative yields a bilinear operation on Te. It turns out that this bilinear operation satisfies the axioms of a Lie bracket, and it is equal to the one defined through left invariant vectorfields.
A vector v in g 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 the exponential function 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. This exponential map is a generalization of the exponential function for real numbers (since R is the Lie algebra of the Lie group of positive real numbers with multiplication) and for matrices (since M(n,R) with the regular commutator is the Lie algebra of the Lie group GL(n,R) of all invertible matrices).
Because of the exponential map, the local structure of every Lie group is determined by its Lie algebra. The global structure however is not; see the table below for examples of different Lie groups sharing the same Lie algebra.
For every finite dimensional real Lie algebra g there is a unique simply connected Lie group G with g as Lie algebra. Moreover every homomorphism between Lie algebra's lifts to a unique homomorphism between the corresponding simply connected Lie Groups.
Examples
| Symbol of Lie group | Description | Remarks | Symbol of Lie algebra | Description |
|---|---|---|---|---|
| Rn | Euclidean space with addition | simply connected | Rn | the Lie bracket is zero |
| R× | nonzero real numbers with multiplication | not connected | R | the Lie bracket is zero |
| R>0 | positive real numbers with multiplication | simply connected | R | the Lie bracket is zero |
| S1 = R/Z | complex numbers of absolute value 1, with multiplication | connected, not simply connected | R | the Lie bracket is zero |
| GL(n,R) | invertible n-by-n real matrices | not connected | M(n,R) | n-by-n matrices, with Lie bracket the commutator |
| GL+(n,R) | n-by-n real matrices with positive determinant | simply connected | M(n,R) | n-by-n matrices, with Lie bracket the commutator |
| SL(n,R) | real matrices with determinant 1 | simply connected | sl(n,R) | square matrices with trace 0, with Lie bracket the commutator |
| O(n) | real orthogonal matrices | ?? | ??, with Lie bracket the commutator | |
| SO(n) | real orthogonal matrices with determinant 1 | so(n) | skew-symmetric square real matrices, with Lie bracket the commutator |
- Please add more entries to the table
Alternative definition
Sometimes, Lie groups are defined as topological manifolds with continuous group operations; this definition is equivalent to our definition given above. 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.