structural:summarization
The exponential map

The exponential map from the Lie algebra M (n; C) {\displaystyle \mathrm {M} (n;\mathbb {C})} of the general linear group G L (n; C) {\displaystyle \mathrm {GL} (n;\mathbb {C})} to G L (n; C) {\displaystyle \mathrm {GL} (n;\mathbb {C})} is defined by the matrix exponential, given by the usual power series:

for matrices X {\displaystyle X}. If G {\displaystyle G} is a closed subgroup of G L (n; C) {\displaystyle \mathrm {GL} (n;\mathbb {C})}, then the exponential map takes the Lie algebra of G {\displaystyle G} into G {\displaystyle G}; thus, we have an exponential map for all matrix groups. Every element of G {\displaystyle G} that is sufficiently close to the identity is the exponential of a matrix in the Lie algebra.

The definition above is easy to use, but it is not defined for Lie groups that are not matrix groups, and it is not clear that the exponential map of a Lie group does not depend on its representation as a matrix group. We can solve both problems using a more abstract definition of the exponential map that works for all Lie groups, as follows.

For each vector X {\displaystyle X} in the Lie algebra g {\displaystyle {\mathfrak {g}}} of G {\displaystyle G} (i.e., the tangent space to G {\displaystyle G} at the identity), one proves that there is a unique one-parameter subgroup c: R → G {\displaystyle c:\mathbb {R} \rightarrow G} such that c ′ (0) = X {\displaystyle c'(0)=X}. Saying that c {\displaystyle c} is a one-parameter subgroup means simply that c {\displaystyle c} is a smooth map into G {\displaystyle G} and that

for all s {\displaystyle s} and t {\displaystyle t}. The operation on the right hand side is the group multiplication in G {\displaystyle G}. The formal similarity of this formula with the one valid for the exponential function justifies the definition

This is called the exponential map, and it maps the Lie algebra g {\displaystyle {\mathfrak {g}}} into the Lie group G {\displaystyle G}. It provides a diffeomorphism between a neighborhood of 0 in g {\displaystyle {\mathfrak {g}}} and a neighborhood of e {\displaystyle e} in G {\displaystyle G}. This exponential map is a generalization of the exponential function for real numbers (because R {\displaystyle \mathbb {R} } is the Lie algebra of the Lie group of positive real numbers with multiplication), for complex numbers (because C {\displaystyle \mathbb {C} } is the Lie algebra of the Lie group of non-zero complex numbers with multiplication) and for matrices (because M (n, R) {\displaystyle M(n,\mathbb {R})} with the regular commutator is the Lie algebra of the Lie group G L (n, R) {\displaystyle \mathrm {GL} (n,\mathbb {R})} of all invertible matrices).

Because the exponential map is surjective on some neighbourhood N {\displaystyle N} of e {\displaystyle e}, it is common to call elements of the Lie algebra infinitesimal generators of the group G {\displaystyle G}. The subgroup of G {\displaystyle G} generated by N {\displaystyle N} is the identity component of G {\displaystyle G}.

The exponential map and the Lie algebra determine the local group structure of every connected Lie group, because of the Baker–Campbell–Hausdorff formula: there exists a neighborhood U {\displaystyle U} of the zero element of g {\displaystyle {\mathfrak {g}}}, such that for X, Y ∈ U {\displaystyle X,Y\in U} we have

where the omitted terms are known and involve Lie brackets of four or more elements. In case X {\displaystyle X} and Y {\displaystyle Y} commute, this formula reduces to the familiar exponential law exp (X) exp (Y) = exp (X + Y) {\displaystyle \exp(X)\exp(Y)=\exp(X+Y)}

The exponential map relates Lie group homomorphisms. That is, if ϕ: G → H {\displaystyle \phi:G\to H} is a Lie group homomorphism and ϕ ∗: g → h {\displaystyle \phi _{*}:{\mathfrak {g}}\to {\mathfrak {h}}} the induced map on the corresponding Lie algebras, then for all x ∈ g {\displaystyle x\in {\mathfrak {g}}} we have

In other words, the following diagram commutes,

(In short, exp is a natural transformation from the functor Lie to the identity functor on the category of Lie groups.)

The exponential map from the Lie algebra to the Lie group is not always onto, even if the group is connected (though it does map onto the Lie group for connected groups that are either compact or nilpotent). For example, the exponential map of SL(2, R) is not surjective. Also, the exponential map is neither surjective nor injective for infinite-dimensional (see below) Lie groups modelled on C Fréchet space, even from arbitrary small neighborhood of 0 to corresponding neighborhood of 1.