In mathematics, a Lie group (IPA liː) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. Lie groups are important in mathematical analysis, physics and geometry because they serve to describe the symmetry of analytical structures. They were introduced by Sophus Lie in 1870 in order to study symmetries of differential equations.
While the Euclidean space Rn is a real Lie group (with ordinary vector addition as the group operation), more typical examples are given by matrix Lie groups, i.e. groups of invertible matrices (under matrix multiplication). For instance, the group SO(3) of all rotations in 3-dimensional space is a matrix Lie group. For a more complete list of examples see the table of Lie groups.
Types of Lie groups
One classifies Lie groups regarding their algebraic properties (simple, semisimple, solvable, nilpotent, abelian), their connectedness (connected or simply connected) and their compactness.
Homomorphisms and isomorphisms
If G and H are Lie groups (both real or both complex), then a Lie-group-homomorphism f : G -> H is a group homomorphism which is also an analytic map. (One can show that it is equivalent to require that f only be continuous.) The composition of two such homomorphisms is again a homomorphism, and the class of all (real or complex) Lie groups, together with these morphisms, forms a category. The two Lie groups are called isomorphic if there exists a bijective homomorphism between them whose inverse is also a homomorphism. Isomorphic Lie groups do not need to be distinguished for all practical purposes; they only differ in the notation of their elements.
The Lie algebra associated to a Lie group
To every Lie group, we can associate a Lie algebra which completely captures the local structure of the group, at least if the Lie group is connected. This is done as follows.
Conventionally, one regards any field X of tangent vectors on a Lie group as a partial differential operator, denoting by Xf the Lie derivative (the directional derivative) of the scalar field f in the direction of X. Then a vector field on a Lie group G is said to be left-invariant if it commutes with left translation, which means the following. Define Lg[f](x) = f(gx) for any analytic function f : G -> F and all g, x in G (here F stands for the field R or C). Then the vector field X
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 F. On a Lie group G, the left-invariant vector fields form a subalgebra, the Lie algebra associated with G, 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
- (x, y) -> xyx−1y−1
on G × G 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 vector fields.
Every 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), for complex numbers (since C is the Lie algebra of the Lie group of non-zero complex 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 the exponential map is surjective on some neighbourhood N of e, it is common to call elements of the Lie algebra infinitesimal generators of the group G. The subgroup of G generated by N will in fact only be the whole group G when G is connected.
The exponential map and the Lie algebra determine the local group structure of every connected Lie group, because of the Campbell-Hausdorff formula: there exists a neighborhood U of the zero element of g, such that for u, v in U we have
- exp(u) exp(v) = exp(u + v + 1/2 [u, v] + 1/12 [[u, v], v] − 1/12 [[u, v], u] − ...)
where the omitted terms are known and involve Lie brackets of four or more elements. In case u and v commute, this formula reduces to the familiar exponential law exp(u) exp(v) = exp(u + v).
Every homomorphism f : G -> H of Lie groups induces a homomorphism between the corresponding Lie algebras g and h. The association G |-> g is a functor.
The global structure of a Lie group is in general not completely determined by its Lie algebra; see the table of Lie groups for examples of different Lie groups sharing the same Lie algebra. We can say however that a connected Lie group is simple, semisimple, solvable, nilpotent, or abelian if and only if its Lie algebra has the corresponding property.
If we require that the Lie group be simply connected, then the global structure is determined by its Lie algebra: for every finite dimensional Lie algebra g over F there is a unique (up to isomorphism) simply connected Lie group G with g as Lie algebra. Moreover every homomorphism between Lie algebras lifts to a unique homomorphism between the corresponding simply connected Lie groups.
Sometimes, real Lie groups are defined as topological manifolds with continuous group operations; this definition is equivalent to our definition given above. This is an interpretation of the content of Hilbert's fifth problem (see Hilbert-Smith conjecture). 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.
Therefore one can also take the definition to use smooth functions. This probably the most common approach now, in textbooks.