Lie Groups and the Exponential Map

A Lie group is a group that is also a smooth manifold, so that the tools of calculus apply to symmetry. This is the setting for every continuous symmetry: rotations, Lorentz transformations, gauge transformations. This page introduces the group–manifold structure and the exponential map that linearises it, handing off the algebraic side to lie-algebras.md and the concrete examples to matrix-groups.md. Manifold prerequisites — manifold, tangent space, smooth map — are taken from topology-manifolds.md.

Definition

A Lie group is a group $G$ that is a smooth ($C^{\infty }$) manifold, such that multiplication $G\times G\to G$, $(g,h)\mapsto gh$, and inversion $G\to G$, $g\mapsto g^{-1}$, are smooth maps. For matrix Lie groups — the only kind we use — "smooth" just means the matrix entries depend smoothly on the parameters, and the group is a closed subgroup of some $GL(n,\mathbb{F})$.

Examples: ${\mathbb{R}}^n$ (translations), $U(1)=S^1$, $SO(n)$, $SU(n)$, $SL(n,\mathbb{R})$, $SL(n,\mathbb{C})$, the Lorentz group ${\mathcal{L}}_{+}^{\uparrow }$, the Poincaré group ${\mathcal{P}}_{+}^{\uparrow }$ — see matrix-groups.md.

The dimension of $G$ is its dimension as a manifold — the number of continuous parameters. A point of subtlety: a Lie group can be disconnected; the analysis here describes the identity component $G^0$ (the piece continuously reachable from $e$). The global component structure is topology-covers.md.

The tangent algebra at the identity

Because a Lie group is homogeneous (left multiplication $L_g:h\mapsto gh$ is a diffeomorphism carrying $e$ to $g$), its geometry is determined by the neighbourhood of the identity. The tangent space there, $$\mathfrak{g}=T_{e}G,$$ is the Lie algebra of $G$ — a vector space of dimension $\dim G$, carrying the bracket developed in lie-algebras.md. A tangent vector at $e$ is the velocity of a curve through the identity; for matrix groups it is a matrix $X$ such that $\gamma (t)=1+tX+O(t^2)$ stays in $G$ to first order.

The exponential map

The exponential map $\exp :\mathfrak{g}\to G$ carries an algebra element to a one-parameter subgroup: $$\exp (tX)\in G,\exp (0)=e,\frac{d}{dt}{|}_{t=0}\exp (tX)=X.$$ The curve $t\mapsto \exp (tX)$ is the unique homomorphism $\mathbb{R}\to G$ with initial velocity $X$; it satisfies $\exp ((s+t)X)=\exp (sX)\exp (tX)$. For matrix groups it is literally the matrix exponential $$\exp (X)=\sum _{n=0}^{\infty }\frac{X^n}{n!}.$$

Key properties.

• $\exp$ is a diffeomorphism from a neighbourhood of $0\in \mathfrak{g}$ onto a neighbourhood of $e\in G$ — so near the identity, group elements are parameterised by algebra elements. This is the precise sense in which "the algebra is the group, linearised."
• On the connected component, every element is a product of exponentials (and for compact groups, a single exponential suffices). Globally $\exp$ need not be surjective or injective — e.g. in $SL(2,\mathbb{R})$ it misses some elements.
• $\exp (X)\exp (Y)=\exp (Z)$ with $Z$ given by the Baker–Campbell–Hausdorff series $Z=X+Y+\frac{1}{2}[X,Y]+\cdots$ — non-commutativity of the group is encoded entirely in the bracket of the algebra (see lie-algebras.md).
• $\det (\exp X)=\exp (\mathrm{tr}X)$ — so $\mathrm{tr}X=0$ (the algebra $\mathfrak{sl}$) exponentiates to $\det =1$ (the group $SL$).

Physics parameterisation

Physicists write near-identity elements with Hermitian generators $T^a$ and real parameters ${\theta }^a$: $$g=\exp \left(i{\theta }^a{T}^a\right)\in G.$$ The factor $i$ makes $T^a$ Hermitian (hence observable-like and giving unitary $g$ when the $T^a$ are Hermitian). The $T^a$ form a basis of $\mathfrak{g}$; their commutators define the structure constants — the entire subject of lie-algebras.md. A finite transformation is recovered from an infinitesimal one by exponentiation, so conserved charges (generators) and finite symmetries (group elements) are two views of the same object — the group-theoretic content of Noether's theorem.

The Lie correspondence

The passage $G⇝\mathfrak{g}$ is close to a dictionary:

Lie's theorems (informal). (i) Every Lie group has a Lie algebra $\mathfrak{g}=T_{e}G$. (ii) A homomorphism of Lie groups induces one of Lie algebras, and near the identity the group homomorphism is determined by the algebra one. (iii) Every finite-dimensional real Lie algebra is the Lie algebra of a unique simply connected Lie group; all connected groups with that algebra are quotients of it by discrete central subgroups.

The failure of (iii) to be an exact bijection — several groups sharing one algebra — is precisely the covering-group phenomenon ($SU(2)$ vs $SO(3)$, $SL(2,\mathbb{C})$ vs the Lorentz group), treated in topology-covers.md. It is why the algebra alone cannot see the difference between a group and its cover, but the global topology can.

References

• Hall, Lie Groups, Lie Algebras, and Representations, Ch. 2–3.
• Warner, Foundations of Differentiable Manifolds and Lie Groups.
• Stillwell, Naive Lie Theory — a gentle route through $\exp$ and one-parameter subgroups.