Every Lie Grouphas an associated Lie algebra whose underlying vector space is the tangent space of G at the identity element, which completely captures the local structure of the group. We can think of elements of the Lie algebra as elements of the group that are "infinitesimally close" to the identity, and the Lie bracket is the commutator of two such infinitesimal elements if G is a matrix group.
The Lie algebra of the vector spaceis justwith the Lie bracket given by
- (In general the Lie bracket of a connected Lie group is always 0 if and only if the Lie group is abelian.)
The Lie algebra of the general linear groupof invertible matrices is the vector spaceof square matrices with the Lie bracket given by
Ifis a closed subgroup ofthen the Lie algebra ofcan be thought of informally as the matricesofsuch thatis inwhereis infinitesimally small so that For example, the orthogonal groupconsists of matriceswithso the Lie algebra consists of the matriceswithwhich is equivalent tobecause
Not all Lie groups can be represented in terms of matrices, so the general definition of the Lie algebra of any Lie group is given by the following steps:
1. Vector fields on any smooth manifoldcan be thought of as derivativesof the ring of smooth functions on the manifold, and therefore form a Lie algebra under the Lie bracketbecause the Lie bracket of any two derivations is a derivation.
2. Ifis any group acting smoothly on the manifoldthen it acts on the vector fields, and the vector space of vector fields fixed by the group is closed under the Lie bracket and therefore also forms a Lie algebra.
3. We apply this construction to the case when the manifoldis the underlying space of a Lie groupwithacting onby left translationsThis shows that the space of left invariant vector fields (vector fields satisfyingfor everywheredenotes the differential of) on a Lie group is a Lie algebra under the Lie bracket of vector fields.
4. Any tangent vector at the identity of a Lie group can be extended to a left invariant vector field by left translating the tangent vector to other points of the manifold. Specifically, the left invariant extension of an elementof the tangent space at the identity is the vector field defined byThis identifies the tangent spaceat the identity with the space of left invariant vector fields, and therefore makes the tangent space at the identity into a Lie algebra, called the Lie algebra of G,
This Lie algebra is finite-dimensional and it has the same dimension as the manifold G. The Lie algebra of G determines G up to "local isomorphism", where two Lie groups are called locally isomorphic if they look the same near the identity element. Problems about Lie groups are often solved by first solving the corresponding problem for the Lie algebras, and the result for groups then usually follows easily. For example, simple Lie groups are usually classified by first classifying the corresponding Lie algebras.
We can also define a Lie algebra structure onusing right invariant vector fields instead of left invariant vector fields. This leads to the same Lie algebra, because the inverse map on G can be used to identify left invariant vector fields with right invariant vector fields, and acts as the inverse on the tangent space
The Lie algebra structure on can also be described as follows: the commutator operation
- onsendstoso its derivative yields a bilinear operation on