A Lie group is simultaneously a smooth manifold and a group. Translation turns tangent data at the identity into global invariant vector fields, so local differential information becomes an algebra that controls much of the global geometry.
Lie groups and invariant fields
Definition. A Lie group is a smooth manifold \(G\) whose multiplication and inversion are smooth. Its Lie algebra is \[ \mathfrak g=T_eG \] with bracket obtained by identifying \(v\in T_eG\) with the left-invariant field \[ X_v(g)=\mathrm{d}(L_g)_ev. \]
Proposition. Left-invariant vector fields are closed under Lie bracket, and evaluation at \(e\) identifies them with \(\mathfrak g\).
Proof. Naturality of the bracket under diffeomorphisms gives \[ (L_g)_*[X,Y]=[(L_g)_*X,(L_g)_*Y]. \] If \(X,Y\) are left invariant, so is \([X,Y]\). A left-invariant field is determined by its value at \(e\), and the formula above constructs one from every tangent vector.
Matrix Lie groups
Example. \[ \begin{aligned} \mathfrak{gl}(n,\mathbb R)&=M_n(\mathbb R),\\ \mathfrak{sl}(n,\mathbb R)&=\{X:\operatorname{tr}X=0\},\\ \mathfrak{so}(n)&=\{X:X^\top=-X\},\\ \mathfrak u(n)&=\{X:X^*=-X\}. \end{aligned} \] For matrix groups the Lie bracket is \([X,Y]=XY-YX\).
Derivation of \(\mathfrak{so}(n)\). If \(A(t)\in O(n)\), \(A(0)=I\), and \(A'(0)=X\), differentiating \(A(t)^\top A(t)=I\) gives \(X^\top+X=0\). Conversely, \(\exp(tX)\in O(n)\) for skew-symmetric \(X\).
The exponential map
Definition. For \(X\in\mathfrak g\), let \(\gamma_X\) be the one-parameter subgroup with \(\dot\gamma_X(0)=X\). Define \[ \exp X=\gamma_X(1). \] For matrix groups this agrees with the matrix exponential.
Theorem. \(\exp((s+t)X)=\exp(sX)\exp(tX)\) and \(\mathrm{d}\exp_0=\operatorname{id}_{\mathfrak g}\). Thus \(\exp\) is a local diffeomorphism near \(0\).
The Baker-Campbell-Hausdorff expansion \[ \log(e^Xe^Y) =X+Y+\frac12[X,Y] +\frac1{12}[X,[X,Y]]+\frac1{12}[Y,[Y,X]]+\cdots \] shows how the bracket records the first failure of exponentials to commute.
Homomorphisms and the adjoint representation
A Lie-group homomorphism \(F:G\to H\) induces a Lie-algebra homomorphism \(\mathrm{d}F_e:\mathfrak g\to\mathfrak h\). Conjugation \(C_g(h)=ghg^{-1}\) defines \[ \operatorname{Ad}_g=\mathrm{d}(C_g)_e\in\operatorname{GL}(\mathfrak g), \] and differentiating \(\operatorname{Ad}\) gives \[ \operatorname{ad}_X(Y)=[X,Y]. \] These representations connect global group symmetry with linear algebra.
Closed subgroups
Theorem (closed subgroup). Every closed subgroup \(H\) of a Lie group \(G\) is an embedded Lie subgroup.
This theorem is highly nontrivial: no manifold structure need be assumed on \(H\). It makes stabilizers of smooth group actions into Lie groups and underlies the homogeneous-space construction \(G/H\).