Lie Algebras & the Classification
Lie Groups & Lie Algebras
A Lie group is a group that is also a smooth manifold — the group operations (multiplication and inversion) are smooth functions. These are the natural language for continuous symmetry, in contrast to the discrete reflection groups of Parts I and II.
The tangent space at the identity element of a Lie group carries a remarkable algebraic structure: the Lie bracket \([x, y]\), which measures the infinitesimal non-commutativity of the group. This tangent space, equipped with the bracket, is called the Lie algebra.
The miracle is that we can almost recover a Lie group from its Lie algebra alone — a finite-dimensional vector space with a bracket operation. "Almost" because we need the group to be connected and simply connected. With those caveats, classifying Lie groups reduces to classifying Lie algebras, which is a problem in linear algebra.
The Killing Form
Every Lie algebra comes with a natural bilinear form, the Killing form: $$B(x, y) = \operatorname{tr}(\operatorname{ad}_x \circ \operatorname{ad}_y)$$ where \(\operatorname{ad}_x(z) = [x, z]\). This form separates Lie algebras into two extreme cases:
Abelian Lie algebras have \([x,y] = 0\) for all \(x,y\), so the Killing form vanishes identically. They correspond to tori \(T^n = S^1 \times \cdots \times S^1\).
Semisimple Lie algebras are those where the Killing form is non-degenerate. For compact semisimple Lie algebras — the ones we care about — the Killing form is negative definite, so \(\langle x, y \rangle = -B(x,y)\) gives an honest inner product. This inner product is what lets us build root lattices inside the Lie algebra.
The Maximal Torus
Inside every compact semisimple Lie group \(G\) sits a maximal torus \(T\) — a torus subgroup not contained in any larger torus. It is essentially unique: any two maximal tori are conjugate to each other. The rank of \(G\) is the dimension of \(T\).
Select a Lie group to see its maximal torus and the lattice hiding inside.
The Lie algebra of the maximal torus, \(\operatorname{Lie}(T)\), is a real vector space isomorphic to \(\mathbb{R}^n\). The Killing form restricts to an inner product on this space. And sitting inside it is the integer lattice \(L\), consisting of all elements \(x\) with \(\exp(2\pi x) = 1\).
The dual lattice \(L^*\) lives in \(\operatorname{Lie}(T)^*\). With the right choice of Lie group (the one with trivial center), this dual lattice is precisely a root lattice in the sense of Part II.
The Weyl Group
Some elements of \(G\) conjugate the maximal torus \(T\) back to itself. Those that act trivially form the normalizer. The quotient is the Weyl group: $$W(G) = \frac{\{g \in G \mid gTg^{-1} = T\}}{\{g \in G \mid gtg^{-1} = t \;\forall\, t \in T\}}$$
The Weyl group acts on \(\operatorname{Lie}(T)\), on the integer lattice \(L\), and on the root lattice \(L^*\). And here is the payoff: the Weyl group is a finite reflection group, generated by reflections through certain vectors in \(L^*\).
Select a rank-2 root system. Click "Apply reflection" to see the Weyl group permute the roots.
The Grand Correspondence
Three wonderful theorems tie everything together. First, the Weyl group \(W(G)\) is a finite reflection group. Second, it is generated by reflections through root vectors, so the root lattice and Weyl group are determined by a Dynkin diagram. Third, the Lie algebra itself is determined by the root lattice and the Weyl group action.
Every compact semisimple Lie algebra decomposes as a direct sum of simple ones, corresponding to disjoint unions of Dynkin diagrams. So the building blocks are classified by the connected Dynkin diagrams — and we know exactly what they are.
The Classical Families
The four infinite series of compact simple Lie algebras all come from rotations in vector spaces over the three associative normed division algebras: the reals \(\mathbb{R}\), the complex numbers \(\mathbb{C}\), and the quaternions \(\mathbb{H}\).
An → \(\mathfrak{su}(n+1)\)
Traceless skew-Hermitian \((n\!+\!1)\times(n\!+\!1)\) complex matrices. The Lie algebra of \(\mathrm{SU}(n\!+\!1)\), the group preserving the inner product on \(\mathbb{C}^{n+1}\).
Bn → \(\mathfrak{so}(2n+1)\)
Traceless skew-symmetric \((2n\!+\!1)\times(2n\!+\!1)\) real matrices. In odd dimensions, every rotation fixes some nonzero vector.
Cn → \(\mathfrak{sp}(n)\)
Traceless skew-Hermitian \(n \times n\) quaternionic matrices. Preserves the inner product on \(\mathbb{H}^n\), with quaternion-linearity.
Dn → \(\mathfrak{so}(2n)\)
Traceless skew-symmetric \(2n\times 2n\) real matrices. In even dimensions, you can find \(n\) orthogonal planes and rotate in each independently.
Slide to change n and watch the dimensions grow.
The Exceptional Five
Beyond the four classical families, exactly five exceptional compact simple Lie algebras exist. They were discovered through the classification itself, not through any pre-existing geometric motivation. Yet they turn out to be deeply connected to a single extraordinary algebraic structure: the octonions \(\mathbb{O}\).
There is a poignant fact here. The constructions of \(\mathfrak{e}_7\) and \(\mathfrak{e}_8\) via octonion tensor products give Riemannian manifolds \((\mathbb{H} \otimes \mathbb{O})\mathrm{P}^2\) and \((\mathbb{O} \otimes \mathbb{O})\mathrm{P}^2\) — but nobody currently knows how to construct these manifolds without first building their isometry groups. The king of simple Lie algebras, \(\mathfrak{e}_8\), remains deeply mysterious.