A Lie group is a group G equipped with the structure of a smooth manifold such that the group operations are smooth maps:
μ:G×G→G,μ(g,h)=gh,ι:G→G,ι(g)=g−1.
For each g∈G, the left translationLg(h)=gh and the right translationRg(h)=hg are diffeomorphisms of G, with inverses Lg−1 and Rg−1. The tangent space at the identity TeG carries a canonical Lie algebra structure, called the Lie algebra of G, and the exponential map relates this infinitesimal structure to local group behavior near e.
Definition. A (smooth) vector field on M is a smooth map X:M→TM such that π∘X=idM. Equivalently, X is a smooth section of the tangent bundle, assigning to each p∈M a tangent vector
Xp∈TpM
(where TpM is the tangent space at p) in a way that is smooth in local coordinates.
A vector field can also be viewed as a derivation on smooth functions: for each X and each f∈C∞(M), one obtains a smooth function X(f)∈C∞(M) defined by differentiating f in the direction X. Using the pairing between tangent and cotangent spaces (see the cotangent bundle), this can be written pointwise as
A Lie bracket on a real vector space g is a bilinear map
[,]:g×g→g
such that:
Alternating / skew-symmetry:[X,X]=0 for all X∈g (equivalently [X,Y]=−[Y,X]).
Jacobi identity: for all X,Y,Z∈g,
[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0.
In differential geometry, there is a canonical Lie bracket on the space of vector fields on a smooth manifoldM: for vector fields X,Y define [X,Y] by its action on smooth functions,
[X,Y](f):=X(Y(f))−Y(X(f)),f∈C∞(M).
This produces another vector field and turns the space of vector fields into a Lie algebra.