Moreover, it is closed under the Lie bracket of vector fields; i.e.,
is a Lie subalgebra of the Lie algebra of all vector fields on G and is called the Lie algebra of G. One can understand this more concretely by identifying the space of left-invariant vector fields with the tangent space at the identity, as follows: Given a left-invariant vector field, one can take its value at the identity, and given a tangent vector at the identity, one can extend it to a left-invariant vector field.
Thus, the Lie algebra can be thought of as the tangent space at the identity and the bracket of X and Y in
can be computed by extending them to left-invariant vector fields, taking the bracket of the vector fields, and then evaluating the result at the identity.
as the Lie algebra of primitive elements of the Hopf algebra of distributions on G with support at the identity element; for this, see #Related constructions below.
Then the Lie algebra of G may be computed as[2][3] For example, one can use the criterion to establish the correspondence for classical compact groups (cf.
is an immersed subgroup of H. If f is surjective, then f is a submersion and if, in addition, G is compact, then f is a principal bundle with the structure group its kernel.
[8] In general, if U is a neighborhood of the identity element in a connected topological group G, then
is the Lie algebra of real square matrices of size n and
In the second part of the correspondence, the assumption that G is simply connected cannot be omitted.
[16] For readers familiar with category theory the correspondence can be summarised as follows: First, the operation of associating to each connected Lie group
from the simply connected covering; its surjectivity corresponds to
Let G be the closed (without taking the closure one can get pathological dense example as in the case of the irrational winding of the torus) subgroup of
be a simply connected covering of G; it is not hard to show that
The preceding can be summarized to saying that there is a canonical bijective correspondence between
and the set of one-parameter subgroups of G.[17] One approach to proving the second part of the Lie group-Lie algebra correspondence (the homomorphisms theorem) is to use the Baker–Campbell–Hausdorff formula, as in Section 5.7 of Hall's book.
indicating other terms expressed as repeated commutators involving X and Y.
The next stage in the argument is to extend f from a local homomorphism to a global one.
The homomorphisms theorem (mentioned above as part of the Lie group-Lie algebra correspondence) then says that if
On the other hand, the group SU(2) is simply connected with Lie algebra isomorphic to that of SO(3), so every representation of the Lie algebra of SO(3) does give rise to a representation of SU(2).
is determined by the group law on G. By Lie's third theorem, there exists a subgroup
for all g in G.[21] Let G be a Lie group acting on a manifold X and Gx the stabilizer of a point x in X.
[23] The kernel of it is a discrete group (since the dimension is zero) called the integer lattice of G and is denoted by
of G; in other words, G fits into the central extension Equivalently, given a Lie algebra
Let G be a connected Lie group with finite center.
It is important to emphasize that the equivalence of the preceding conditions holds only under the assumption that G has finite center.
Thus, for example, if G is compact with finite center, the universal cover
The last three conditions above are purely Lie algebraic in nature.
and the right-hand side is the de Rham cohomology of G. (Roughly, this is a consequence of the fact that any differential form on G can be made left invariant by the averaging argument.)
be the algebra of distributions on G with support at the identity element with the multiplication given by convolution.