The flat shadow of a curved group
In the previous guide you met a Lie group G as a smooth manifold that is also a group, with smooth multiplication and inversion, and you saw the classical matrix groups GL(n), SL(n), O(n), SU(n) as the working examples. Such a group is a genuinely curved object — SU(2) is diffeomorphic to the 3-sphere S^3, which has no flat global coordinates. The central discovery of this rung is that almost everything you want to know about G is already encoded in a single tangent space, the tangent space at the identity element e, and that this tangent space is not just a vector space but carries an extra multiplication-like operation called a bracket.
Why the identity, and not some other point? Because a group is homogeneous: for any g in G the map L_g that sends x to g x — left translation — is a diffeomorphism of G that carries e to g. So the group looks the same near every point, and the identity is merely the most convenient base point to stand on. The vector space T_e G, the tangent space at e, is the raw material. Adding the bracket to it turns T_e G into the Lie algebra of G, written with a fraktur lowercase letter, g (read "gothic g"). The whole guide is about where that bracket comes from honestly, not by decree.
Spreading one tangent vector across the whole group
Here is the trick that makes everything work. Pick a single tangent vector v in T_e G. Using left translation we can copy it to every point g of the group at once: at g, declare the vector to be the pushforward of v by L_g, that is (dL_g)_e applied to v. This gives a vector field X on all of G, smooth because multiplication is smooth, and X is called left-invariant precisely because it is unchanged when you push it forward by any left translation: (L_h) carries X to itself for every h in G.
This sets up a perfect dictionary. A left-invariant vector field is determined everywhere by its single value at e, and conversely any vector in T_e G grows into exactly one left-invariant field. So the vector space of left-invariant vector fields on G is canonically the same as T_e G — same dimension, same linear structure. The payoff is that we now have two faces of the same object: T_e G is finite-dimensional and easy to picture, while the left-invariant fields are global geometric objects you can differentiate against. The bracket will be defined on the global side, then read back to T_e G.
v in T_e G <----> X^v a left-invariant vector field on G X^v(g) := (dL_g)_e (v) (push v forward by left translation) invariance: (L_h)_* X^v = X^v for every h in G recover the vector: X^v(e) = v
Where the bracket comes from
Recall from the manifolds rung that any two smooth vector fields X and Y on a manifold have a Lie bracket [X, Y], another vector field that measures the failure of their flows to commute. Concretely, if you flow a little along X, then along Y, then back along X, then back along Y, you do not in general return to where you started; the bracket [X, Y] is the leading-order vector pointing in the direction of that discrepancy. It also satisfies [X, Y] = XY - YX when you treat the fields as differential operators acting on functions.
Now the decisive fact, and the heart of the whole subject: the bracket of two left-invariant vector fields is again left-invariant. This is not obvious, but it follows from the fact that the bracket is natural under any diffeomorphism — pushing forward commutes with taking brackets — and left translations are diffeomorphisms under which X and Y are each fixed. So if X and Y are left-invariant, so is [X, Y]. Therefore the bracket stays inside the space of left-invariant fields, which we identified with T_e G. Reading the result back to e gives a bilinear operation on T_e G, and that operation is the Lie bracket of the Lie algebra g.
What the bracket obeys, and what it means
A Lie algebra is, abstractly, a finite-dimensional vector space with a bracket [.,.] that is bilinear, antisymmetric ([X, X] = 0), and satisfies the Jacobi identity [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0. The Jacobi identity is not decoration: it is exactly the infinitesimal residue of the associativity of the group multiplication. So associativity of G — a global, curved property — survives the collapse to T_e G as this one clean algebraic law. That is the precise sense in which the Lie algebra remembers the group it came from.
For matrix groups the bracket has a delightfully concrete face. If G sits inside GL(n) as matrices, then T_e G is a space of matrices, and the Lie bracket is simply the commutator [A, B] = AB - BA, ordinary matrix products subtracted. This is no accident: it is the same XY - YX formula, computed in the linear coordinates that matrix entries provide. So for example the Lie algebra of SU(2) is the space of 2x2 skew-Hermitian traceless matrices, a real 3-dimensional space, and its bracket is matrix commutator — which, in a chosen basis, reproduces the cross-product structure of R^3.
Make that example fully explicit, because it repays the effort. The algebra su(2) is the set of 2x2 complex matrices A with A* = -A and trace A = 0, a real 3-dimensional space. Take the basis e_1 = (i/2) sigma_1, e_2 = (i/2) sigma_2, e_3 = (i/2) sigma_3, where sigma_1, sigma_2, sigma_3 are the Pauli matrices. Computing commutators by hand gives [e_1, e_2] = e_3, [e_2, e_3] = e_1, [e_3, e_1] = e_2 — exactly the relations the cross product obeys on R^3. The structure constants here are the Levi-Civita symbol, and you have just met, in miniature, the algebra that governs angular momentum in quantum mechanics.
Why a finite list of numbers controls the group
Fix a basis e_1, ..., e_n of g. Then every bracket [e_i, e_j] is itself a vector, so it expands as a sum of c^k_ij e_k for some real numbers c^k_ij, the structure constants. Antisymmetry and the Jacobi identity become polynomial equations on these constants, and once they are fixed the entire bracket — and hence the whole infinitesimal structure of the group — is determined. A continuous, infinite, curved object has been compressed to a finite array of numbers. This is the practical miracle that makes Lie theory computable.
- Realize G inside GL(n) and identify g = T_e G as a concrete space of matrices.
- Choose a basis e_1, ..., e_n of that matrix space.
- Compute each commutator [e_i, e_j] = e_i e_j - e_j e_i as a matrix.
- Re-express that matrix in the basis to read off the structure constants c^k_ij.
- Check antisymmetry and the Jacobi identity hold — they must, since the commutator always obeys them.
The deep claim that justifies all this effort is the subject of Lie's theorems: a connected, simply connected Lie group is determined up to isomorphism by its Lie algebra, and a homomorphism of Lie algebras integrates to a homomorphism of such groups. We will not prove these here — that is genuinely a course, and the proof leans on the one-parameter subgroups and the exponential map of the next guide, plus the Frobenius integrability theorem. But the upshot is honest and large: at the infinitesimal level, the linear, finite, computable Lie algebra captures the group almost completely.
What this unlocks next
You now hold the linearization of a Lie group: a flat vector space g = T_e G carrying a bracket that remembers associativity through the Jacobi identity. Everything ahead in this rung is the round trip between g and G. The next guide builds the bridge back upward — the exponential map exp: g -> G, whose images trace out one-parameter subgroups and turn a tangent vector into an actual curve in the group. After that, the adjoint representation will show g acting on itself, the Killing form will let us measure it, and the classification of semisimple Lie algebras by roots will reveal how rigid these objects truly are.