The Adjoint Representation: A Lie Algebra Acting on Itself

Precise statement: two maps and one differential

Let 𝔤 be a Lie algebra over a field k with bracket [·,·]. The adjoint representation is the linear map ad: 𝔤 → 𝔤𝔩(𝔤) = End(𝔤), X ↦ ad_X, where ad_X(Y) = [X, Y]. The claim is that ad is a homomorphism of Lie algebras: ad_[X,Y] = [ad_X, ad_Y] = ad_X ad_Y − ad_Y ad_X, where the bracket on the right is the commutator of endomorphisms.

For a Lie group G with Lie algebra 𝔤 ≅ T_e G, define c_g(h) = g h g⁻¹ (inner automorphism) and set Ad_g = d(c_g)_e ∈ GL(𝔤). Then Ad: G → GL(𝔤) is a smooth group homomorphism (a representation on 𝔤), and its differential at the identity is exactly ad: d(Ad)_e = ad. Equivalently, for all X ∈ 𝔤 one has the fundamental relation Ad(exp X) = exp(ad_X), an equality of operators on 𝔤. For a matrix Lie group these specialize to Ad_g(Y) = gYg⁻¹ and ad_X(Y) = XY − YX.

The picture: conjugation seen from the tangent space

The intuition is conjugation, linearized. The map c_g measures how far G is from abelian: if G were commutative, c_g would be the identity for every g. Ad_g captures the infinitesimal distortion conjugation induces on tangent vectors at e. So Ad packages the entire family of inner automorphisms into a single linear action on the fixed vector space 𝔤.

Differentiating once more in g gives ad. Concretely, ad_X is the infinitesimal generator of the flow Y ↦ Ad(exp(tX))Y, so d/dt|₀ Ad(exp tX)Y = [X, Y]. Thus ad_X is the Lie derivative of the vector field along the one-parameter subgroup exp(tX): the bracket [X,Y] is exactly the failure of the flows of X and Y to commute, to first order. The adjoint representation is therefore the algebraic shadow of curvature — it is how the group 'twists' its own tangent directions into one another.

Key idea: the Jacobi identity IS a representation

The proof that ad is a homomorphism is the Jacobi identity in disguise, and this is the mechanism worth internalizing. We must show ad_[X,Y] = ad_X ad_Y − ad_Y ad_X. Apply both sides to an arbitrary Z:

Left side: ad_[X,Y](Z) = [[X,Y], Z].
Right side: ad_X ad_Y(Z) − ad_Y ad_X(Z) = [X,[Y,Z]] − [Y,[X,Z]].

These agree precisely when [[X,Y],Z] = [X,[Y,Z]] − [Y,[X,Z]], which, rearranged using antisymmetry, is the Jacobi identity [X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0. So ad being a representation is logically equivalent to Jacobi — no computation beyond bookkeeping is needed.

For the group side, Ad_g Ad_h = d(c_g)_e d(c_h)_e = d(c_g ∘ c_h)_e = d(c_{gh})_e = Ad_{gh} by the chain rule and c_g ∘ c_h = c_{gh}. Differentiating the identity Ad(exp X) = exp(ad_X) at X = 0 in the direction X recovers d(Ad)_e = ad.

Worked example: 𝔰𝔲(2) and 𝔰𝔬(3)

Take 𝔤 = 𝔰𝔲(2) with basis X₁, X₂, X₃ satisfying [Xᵢ, Xⱼ] = ε_{ijk} Xₖ (the structure constants are the Levi-Civita symbol). Compute ad_{X₁} in this basis: ad_{X₁}(X₂) = [X₁,X₂] = X₃ and ad_{X₁}(X₃) = [X₁,X₃] = −X₂, while ad_{X₁}(X₁)=0. So the matrix of ad_{X₁} is the 3×3 generator of rotations about the first axis — the adjoint representation of 𝔰𝔲(2) is 𝔰𝔬(3) acting on ℝ³.

Exponentiating, Ad(exp(θX₁)) = exp(θ ad_{X₁}) is rotation by θ about the X₁-axis. This is the concrete face of the famous 2-to-1 cover SU(2) → SO(3): the group Ad(SU(2)) = SO(3), and the kernel of Ad is the center {±I}. The adjoint map literally exhibits SO(3) as SU(2) modulo its center, and shows why a 720° rotation in SU(2) descends to a 360° rotation in SO(3).

Where hypotheses bite: center, semisimplicity, and the Killing form

Faithfulness of ad requires a trivial center: ker(ad) = 𝔷(𝔤) = {X : [X,Y]=0 ∀Y}. For abelian 𝔤 (e.g. ℝⁿ with zero bracket) ad ≡ 0, so the adjoint representation is useless — it forgets everything. Thus 'a Lie algebra acting on itself' is informative exactly to the extent that 𝔤 is non-abelian. At the group level ker(Ad) = Z(G), the full center, for connected G (the differential statement ker(ad) = 𝔷(𝔤) = Lie(Z(G)) only sees the identity component Z(G)°, but Ad itself kills the whole center, including its finite part), so Ad detects G only up to its center.

The payoff appears under semisimplicity. Cartan's criterion: 𝔤 is semisimple iff the Killing form κ(X,Y) = tr(ad_X ad_Y) is non-degenerate. Here the invariance κ(ad_Z X, Y) + κ(X, ad_Z Y) = 0 (ad-invariance) is what makes κ a genuine tool. Drop semisimplicity and κ degenerates — its radical is a solvable ideal — and root-space theory collapses. Connections: the whole Cartan–Killing classification, compactness (κ negative-definite ⟺ compact semisimple), and Whitehead's lemmas / Lie algebra cohomology all hinge on this form built from ad.

Why it matters: roots, curvature, and BCH

The adjoint representation is the engine of semisimple structure theory. Fix a Cartan subalgebra 𝔥 ⊂ 𝔤; the operators {ad_H : H ∈ 𝔥} commute and are simultaneously diagonalizable, giving the root space decomposition 𝔤 = 𝔥 ⊕ ⨁_{α∈Φ} 𝔤_α, where 𝔤_α = {X : ad_H X = α(H)X ∀H}. The roots α are eigenvalues of the adjoint action; the entire A–D–E classification of simple Lie algebras is read off from how these root vectors bracket, i.e. from ad restricted to root spaces.

Beyond classification, Ad(exp X) = exp(ad_X) is the algebraic core of the Baker–Campbell–Hausdorff formula exp X exp Y = exp(X + Y + ½[X,Y] + …), whose higher terms are iterated ad's. In differential geometry, ad governs the curvature of bi-invariant metrics (R(X,Y)Z = −¼[[X,Y],Z]) and the holonomy of principal bundles. In physics it is the gauge field's transformation law: the field strength lives in the adjoint. It is, in short, the representation you cannot avoid.

The adjoint representation at the algebra level (ad) versus the group level (Ad), and the identities linking them.

Feature	ad: 𝔤 → 𝔤𝔩(𝔤)	Ad: G → GL(𝔤)
Formula	ad_X(Y) = [X, Y]	Ad_g(Y) = d(c_g)_e(Y), c_g(h)=ghg⁻¹
For matrix groups	ad_X(Y) = XY − YX	Ad_g(Y) = g Y g⁻¹
Kernel	center 𝔷(𝔤) = {X : [X,·]=0}	Z(G) for connected G; the full center
Linking identity	d(Ad)_e = ad	Ad(exp X) = exp(ad_X)
Invariant form	Killing κ(X,Y)=tr(ad_X ad_Y)	κ(Ad_g X, Ad_g Y) = κ(X,Y)
Faithful when	𝔤 has trivial center	G connected, semisimple, centerless