One-parameter subgroups as velocity
A one-parameter subgroup is a smooth path g(t) = exp(tA) through the identity satisfying g(s+t) = g(s)g(t). Take its derivative at t = 0: d/dt exp(tA) at t=0 = A. So A is the velocity vector of the curve as it leaves the identity. Every direction you can flow away from I is some matrix A, and exp turns that direction back into the flow.
The condition g(s+t) = g(s)g(t) says the curve turns addition of times into multiplication in the group — a homomorphism from the real line into the Lie group. Conversely, every such smooth curve through the identity is exp(tA) for a unique A. So one-parameter subgroups and their generators A correspond exactly, the first concrete dictionary between the group and its tangent directions.
The Lie algebra as tangent space
Collect all those velocity vectors and you get the Lie algebra of the group: the tangent space at the identity, written with a lowercase fraktur name (so(n), su(n), and so on). It is a vector space — closed under addition and scaling — far flatter and easier than the curved group itself. Differentiating each group's defining equation reveals its algebra.
- Write g(t) = exp(tA) and substitute it into the group's defining condition (e.g. Q^T Q = I).
- Differentiate at t = 0 using d/dt exp(tA) = A and the product rule.
- Read off the linear condition on A: that condition defines the Lie algebra.
# Differentiate the orthogonal condition for SO(n):
( exp(tA) )^T ( exp(tA) ) = I for all t
exp(tA^T) exp(tA) = I
d/dt at t=0: A^T + A = 0 => A^T = -A
# so the Lie algebra so(n) = { antisymmetric matrices }.
# Same method gives:
su(n) = { A : A* = -A (anti-Hermitian), trace A = 0 }
sl(n) = { A : trace A = 0 } (from det exp(tA) = e^{t*traceA} = 1)Generators: a tiny basis for a whole group
Because the Lie algebra is a vector space, a basis for it gives a few infinitesimal generators from which the whole group is built by exp. For SO(3) the algebra so(3) is 3-dimensional, with three generators Lx, Ly, Lz — one per rotation axis. exp(theta * Lz) is a rotation about the z-axis by theta. Three small matrices encode every spatial rotation.
Why work in the algebra at all
The Lie algebra is a linearization of the group — it trades curved, non-commutative geometry for a flat vector space you already know how to handle from Volume I. Adding, scaling, and finding a basis all work. But one piece of the group's structure is invisible to plain addition: how generators fail to commute. Capturing that is the job of the bracket, and the payoff of the whole track.