Lie groups and Lie algebras

A Lie group is both a smooth manifold and a group, with smooth multiplication and inversion. Rotations are a concrete example: composing two rotations gives another, the identity does nothing, and every rotation can be undone. For a matrix group, a smooth path through the identity has a matrix derivative. These tangent directions form its Lie algebra, a vector space equipped with a bracket. For matrix groups the bracket is the commutator AB−BA.

The matrix exponential converts an infinitesimal generator into a one-parameter subgroup. For fixed A, exp(tA)exp(sA)=exp((t+s)A). When A and B differ, this simplification generally fails because they need not commute. For finite-dimensional Lie groups the exponential is a local diffeomorphism near zero and the identity, but it need not cover the entire group globally. Moving from local coordinates to global symmetry requires care about connectivity and topology.

Perform a small A motion, then a small B motion, then undo each in reverse individual directions. The remaining transformation starts at second order and is governed by [A,B]. In three-dimensional rotations, rotations around different axes fail to commute, so the rotation algebra is non-Abelian. In the planar rotation group SO(2), the algebra is one-dimensional and the bracket vanishes. Noncommutativity is a structural statement about composition, not simply the presence of complex numbers in a representation.

Choose basis generators T_a and express their brackets as linear combinations of the same generators. The coefficients are structure constants in that basis; they change under a basis change even though the abstract algebra remains the same. Physicists often use Hermitian generators with an extra factor i in the bracket, while real matrix-group conventions often use anti-Hermitian generators. State the convention before comparing signs in angular-momentum or gauge-field equations.

SO(3) and SU(2) have isomorphic real Lie algebras but different global topology. SU(2) is a double cover of SO(3): two SU(2) elements map to each spatial rotation. The infinitesimal commutation relations therefore agree while the global identification of transformations differs. A simply connected Lie group integrates a finite-dimensional real Lie algebra uniquely up to isomorphism, but other connected groups with the same algebra can be obtained through suitable discrete central quotients.

A representation tells us how a group acts linearly on a chosen vector space. Its differential yields a Lie algebra representation, but integrating an algebra representation to a particular global group can impose extra restrictions. Spin-half representations are representations of SU(2) and do not descend to ordinary single-valued representations of SO(3). This is why identifying local generators is only part of specifying a quantum symmetry. Gauge theories likewise require a global group and matter representations in addition to a list of local structure constants.