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A Lie group is a smooth manifold that is also a group, with smooth multiplication and inversion. Its Lie algebra — the tangent space at the identity equipped with the Lie bracket — captures the infinitesimal structure of the group. The exponential map connects the Lie algebra to the group, converting algebra problems to geometry and vice versa. Lie groups are the mathematical language of continuous symmetry, appearing as isometry groups of Riemannian manifolds, structure groups of bundles, and gauge groups in physics.
A Lie group is a group that is simultaneously a smooth manifold, with the group operations (multiplication μ : G × G → G and inversion ι : G → G) being smooth maps. The classical examples are matrix groups: GL(n, ℝ) (invertible matrices), O(n) (orthogonal matrices), SO(n) (rotations), SL(n) (determinant-1 matrices), U(n) and SU(n) (unitary matrices). These are all smooth submanifolds of the space of matrices, and the group operations are restrictions of polynomial (hence smooth) maps.
The Lie algebra 𝔤 of G is the tangent space at the identity TeG, equipped with the Lie bracket inherited from left-invariant vector fields. A vector X ∈ TeG extends uniquely to a left-invariant vector field X̃ on G (by left-translating: X̃_g = dL_g(X)). The bracket [X, Y] is defined as the bracket of the corresponding left-invariant fields: [X, Y] = [X̃, Ỹ]_e. For matrix groups, this bracket is the matrix commutator [A, B] = AB - BA. The Lie algebra is a finite-dimensional vector space with a bilinear, antisymmetric bracket satisfying the Jacobi identity.
The exponential map exp : 𝔤 → G sends X to the time-1 flow of the left-invariant vector field X̃. Equivalently, exp(X) = γ_X(1) where γ_X is the unique one-parameter subgroup with γ_X'(0) = X. For matrix groups, this is the matrix exponential exp(A) = I + A + A²/2! + .... The exponential map is a local diffeomorphism near 0 ∈ 𝔤 (by the inverse function theorem), providing coordinates on a neighborhood of the identity. The Baker-Campbell-Hausdorff formula exp(X)exp(Y) = exp(X + Y + ½[X,Y] + ...) shows how the Lie bracket controls the group multiplication to higher order.
Lie groups pervade differential geometry as symmetry groups. The isometry group of a Riemannian manifold is a Lie group (the Myers-Steenrod theorem). The structure group of a vector bundle or principal bundle is a Lie group. Gauge theories in physics are built on Lie groups. The representation theory of Lie groups — studying homomorphisms from G to GL(V) — is the mathematical backbone of quantum mechanics and particle physics. The classification of simple Lie algebras (Killing, Cartan) is one of the great achievements of 19th-century mathematics, organizing all possible continuous symmetries into families (A_n, B_n, C_n, D_n) and exceptional cases (G₂, F₄, E₆, E₇, E₈).