Part 1 of 15. A vector space with a bilinear, antisymmetric, Jacobi-respecting "bracket" operation. That four-line definition captures the infinitesimal shadow of every continuous symmetry in mathematics, and everything in this series unfolds from it.
This series walks past the specimens in a particular museum: the complete list of finite-dimensional simple Lie algebras over ℂ. Up to isomorphism there are nine species. Four come in infinite families: the classical types A, B, C, D. Five are singular exceptions: E6, E7, E8, F4, and G2. The Cartan–Killing classification theorem says every finite-dimensional simple Lie algebra belongs to exactly one of these. To read that list we first need to agree on what a Lie algebra is, and on what "simple" means for one.
This explainer gives the definition, grounds it in three canonical examples, and introduces the two ideas the classification uses as levers: ideals and the adjoint representation.
The canonical source of Lie algebras is a Lie group, a group whose underlying set is also a smooth manifold, with smooth multiplication and inversion. The rotation group SO(3) is the standard example: its elements are 3×3 orthogonal matrices with determinant 1, forming a 3-dimensional smooth surface inside the 9-dimensional space of all 3×3 matrices. SO(3) is continuous in a way finite groups like S5 are not; you can slide along a smooth path of rotations.
Near the identity, a Lie group looks like a flat vector space: its tangent space at the identity. For SO(3), this tangent space is 3-dimensional. It is the space of infinitesimal rotations, with each vector identified with an axis-and-angular-velocity pair. Euler's rotation theorem says every finite rotation has an axis; the small-rotation limit produces a vector along that axis with magnitude equal to the angular speed.
A vector space alone throws away most of the group's information. The Lie bracket is a second operation that records, on the tangent space, exactly what is needed to reconstruct the group locally. Two infinitesimal rotations compose into a finite rotation whose second-order correction is captured by their bracket. The Lie algebra is the tangent space plus this bracket.
The linearisation trick. A curved group gets flattened into a vector space near the identity, with all the curvature pushed into a single extra operation, the bracket. The vector space is just ℝn. The bracket is where the structure lives.
A Lie algebra is a vector space 𝔤 over a field (we work over ℂ throughout the series) equipped with a binary operation [·, ·]: 𝔤 × 𝔤 → 𝔤 obeying three laws:
There is no fourth law. The Lie bracket is not required to be associative, and in every example we care about it is non-associative. The Jacobi identity serves as a weakened, asymmetric replacement for associativity that is compatible with antisymmetry.
One consequence of bilinearity plus antisymmetry: every element brackets with itself to give zero. Expanding [x + y, x + y] and cancelling yields [x, x] = 0 for every x ∈ 𝔤. The bracket is alternating, a slightly stronger condition than antisymmetric (over fields of characteristic 2 the two are not equivalent, but ℂ has characteristic 0 so there is no distinction here).
The prototypical example: take 𝔤 to be sl2(ℂ), the space of 2×2 complex matrices with trace zero, with bracket the matrix commutator [X, Y] = XY − YX. The commutator of any two matrices is trace zero (because tr(XY) = tr(YX)), so the operation stays inside sl2. Bilinearity follows from distributivity, antisymmetry from the definition. The Jacobi identity can be checked by grinding through the six terms, which is a good thing to do exactly once.
sl2(ℂ) is three-dimensional as a complex vector space, spanned by the canonical basis
with structure constants
These three relations encode everything about sl2. The bracket of any two elements of sl2 can be computed from them using bilinearity.
Figure 1. Live bracket calculator for sl2(ℂ). Pick coefficients of X and Y in the (H, E, F) basis using the sliders. The two input matrices and their commutator [X, Y] = XY − YX are displayed on the right. The commutator always has trace zero; it stays inside sl2. Swap the sliders of X and Y to see the bracket flip sign, just as antisymmetry requires. Set X = Y to get the zero matrix, just as the alternating property requires.
Three constructions of Lie algebras turn up so often that they deserve names.
gln(ℂ), the space of all n×n complex matrices with the commutator bracket [X, Y] = XY − YX, is a Lie algebra of complex dimension n2. Its special linear sibling sln(ℂ) is the trace-zero subspace, of dimension n2 − 1. The commutator automatically lands in sln whenever the operands do, because tr(XY − YX) = 0 identically. sl2 we have already met; sln for n ≥ 2 turns out to be the entire A-type family An−1 in the classification.
so(n) is the space of n×n skew-symmetric real matrices, i.e. matrices A with AT = −A. A skew-symmetric matrix has zero diagonal, so so(n) is (n2 − n)/2-dimensional. Under the commutator bracket, so(n) is a Lie algebra. It is the tangent space at the identity of the orthogonal group SO(n), and its elements generate the rotations of ℝn.
The smallest interesting case is so(3), which is 3-dimensional (a 3×3 skew-symmetric matrix has three free entries above the diagonal). Write the general element as
and you can check directly that the bracket of two such matrices satisfies
where × is the ordinary 3-dimensional cross product. So so(3) and ℝ3 with the cross product are the same Lie algebra. Under v ↦ Av, matrix commutation on so(3) corresponds to cross multiplication on ℝ3. The 3-dimensional cross product is not a generic way to multiply vectors; it exists only because ℝ3 happens to be a Lie algebra.
Figure 2. so(3) ≅ (ℝ3, ×). Two input vectors u and v, and their bracket u × v, shown in an isometric 3D projection. Swap u and v from the presets to see u × v flip sign. Setting u = v gives the zero vector. The cross product magnitude is |u||v|sin θ, so parallel inputs also bracket to zero.
The third source, referenced but not used here, is smooth vector fields on a manifold. Each smooth vector field on M acts as a first-order differential operator on smooth functions. The commutator [X, Y] = XY − YX of two such operators is again a first-order differential operator, a new vector field. The space of all smooth vector fields on M, under this bracket, is a (generally infinite-dimensional) Lie algebra. Restricting to left-invariant vector fields on a Lie group G recovers the finite-dimensional Lie algebra of G.
Commutators and composition order. In all three examples the bracket asks "does the order of two operations matter?" [X, Y] = XY − YX measures the failure of two matrices (or rotations, or vector fields) to commute. Lie algebras where every bracket vanishes are called abelian; they carry no structure beyond being vector spaces, so the classification focuses on the non-abelian case.
What are the substructures of a Lie algebra? Two kinds, and the difference between them is the whole story of the classification.
A subalgebra 𝔥 ⊂ 𝔤 is a vector subspace closed under the bracket: [x, y] ∈ 𝔥 whenever x, y ∈ 𝔥. It is then a Lie algebra in its own right. Diagonal matrices inside sln form a subalgebra (abelian, because they commute); upper-triangular matrices form another; sl2 embedded as the top-left 2×2 block of sl3 is a third.
An ideal 𝔦 ⊂ 𝔤 has a stronger closure property: it absorbs brackets with anything in the ambient algebra, [x, y] ∈ 𝔦 for every x ∈ 𝔤 and y ∈ 𝔦. Every ideal is a subalgebra, but most subalgebras are not ideals. Ideals are the Lie-algebra analogue of normal subgroups: 𝔤/𝔦 inherits a Lie algebra structure exactly when 𝔦 is an ideal.
Every Lie algebra has two trivial ideals: {0} and 𝔤 itself. A Lie algebra is simple if these are the only ideals and 𝔤 is non-abelian. The non-abelian condition excludes the 1-dimensional case ℂ, which has no non-trivial ideals for dimension reasons but would be a degenerate entry on the classification list.
Simple Lie algebras are the indivisible atoms of continuous symmetry, and classifying them is analogous to classifying primes in ℤ or finite simple groups. Every finite-dimensional semisimple Lie algebra over ℂ is a direct sum of simple Lie algebras, unique up to permutation (Weyl's theorem on complete reducibility). Classifying semisimple Lie algebras reduces completely to classifying simple ones.
The construction that drives the classification proof is the adjoint representation. For any x ∈ 𝔤, define the linear operator adx: 𝔤 → 𝔤 by
adx takes y to its bracket with x. Bilinearity makes it linear, so each element of the algebra produces an endomorphism of the algebra itself, and x ↦ adx is a map 𝔤 → End(𝔤).
The Jacobi identity, rewritten in terms of the adjoint, becomes
which says adx is a derivation of the bracket, distributing over it the way an ordinary derivative distributes over a product. Moreover, x ↦ adx itself preserves the Lie bracket:
where the left bracket is the Lie bracket of 𝔤 and the right is the commutator on End(𝔤). So x ↦ adx is a Lie algebra homomorphism, and its image (the inner derivations of 𝔤) is isomorphic to 𝔤 itself whenever the centre of 𝔤 is zero, which it always is for simple Lie algebras.
The adjoint representation lets us study 𝔤 using linear algebra on 𝔤 itself. What are the eigenvalues of adx, and how does 𝔤 decompose into eigenspaces? The answer, for well-chosen x, is the root space decomposition, the central tool for classifying simple Lie algebras. The next explainer defines the Cartan subalgebra as the best collection of x's to diagonalise simultaneously, and the two after that use the simultaneous eigenspaces as the atoms of the classification.
Next we pick a maximal set of elements of 𝔤 whose adjoint operators commute: the Cartan subalgebra. Then we diagonalise them simultaneously. The eigenspaces are the root spaces and the eigenvalues are the roots. From there on, the classification is geometry.