Root system

Intuition [Beginner]

A root system is a symmetric arrangement of vectors in Euclidean space that encodes the structure of a Lie algebra. Think of it as the skeleton: the roots are special directions along which the Lie algebra expands and contracts, and their angles and lengths determine the entire algebra.

The defining feature is reflection symmetry. For each root $\alpha$, the whole configuration is unchanged if you reflect every vector across the hyperplane perpendicular to $\alpha$. If $\alpha$ is a root, then $-\alpha$ is also a root. The roots always come in opposite pairs.

For the Lie algebra $\mathfrak{sl}(3,\mathbb{C})$ of traceless $3\times 3$ matrices, the root system lives in a $2$-dimensional plane and consists of six vectors forming a regular hexagon. The angles between adjacent roots are all $60$ degrees.

Visual [Beginner]

Worked example [Beginner]

Take the root system $A_2$ in the plane. Choose coordinates so the two simple roots are ${\alpha }_1=(1,0)$ and ${\alpha }_2=(-1/2,\sqrt{3}/2)$. The angle between them is $120$ degrees, and they have equal length.

The full set of six roots is ${\alpha }_1$, ${\alpha }_2$, ${\alpha }_1+{\alpha }_2$, $-{\alpha }_1$, $-{\alpha }_2$, and $-{\alpha }_1-{\alpha }_2$. The roots ${\alpha }_1$ and ${\alpha }_2$ are the simple roots; all positive roots (${\alpha }_1$, ${\alpha }_2$, ${\alpha }_1+{\alpha }_2$) are nonnegative combinations of them.

Reflecting ${\alpha }_1$ across the line perpendicular to ${\alpha }_2$ gives ${\alpha }_1+{\alpha }_2$. Reflecting again gives back ${\alpha }_2$. These reflections generate the Weyl group, which for $A_2$ is the symmetry group of the hexagon — six elements, isomorphic to the permutations of three objects.

Check your understanding [Beginner]

Formal definition [Intermediate+]

A root system $\Phi$ in a Euclidean space $E$ with inner product $(\cdot ,\cdot )$ is a finite set of nonzero vectors satisfying four axioms:

(R1) $\Phi$ spans $E$.

(R2) For each $\alpha \in \Phi$, the reflection $s_{\alpha }(v)=v-\frac{2(v,\alpha )}{(\alpha ,\alpha )}\alpha$ preserves $\Phi$.

(R3) For all $\alpha ,\beta \in \Phi$, the quantity $\frac{2(\beta ,\alpha )}{(\alpha ,\alpha )}$ is an integer.

(R4) The only scalar multiples of $\alpha \in \Phi$ that belong to $\Phi$ are $\alpha$ and $-\alpha$.

A root system is called reduced when (R4) holds; some authors include this condition in (R2). ^{[Bourbaki Ch VI; Humphreys §8-10; Serre]}

A set of simple roots $\Delta \subset \Phi$ is a basis of $E$ such that every root is a linear combination of elements of $\Delta$ with all coefficients nonnegative or all coefficients nonpositive. Roots with nonnegative coefficients are positive roots; the others are negative roots. The Weyl group $W$ is the finite group generated by the reflections $\{s_{\alpha }:\alpha \in \Phi \}$.

Key theorem with proof [Intermediate+]

Theorem. The Weyl group $W$ of a root system $\Phi$ is finite, and $W$ acts faithfully on $\Phi$.

Proof. Each reflection $s_{\alpha }$ permutes the finite set $\Phi$ by axiom (R2). Therefore $W$ embeds into the symmetric group on $|\Phi |$ elements, which is finite. Faithfulness holds because any element $w\in W$ fixing every root must fix the span of $\Phi$, which is all of $E$ by axiom (R1). A linear map fixing a spanning set is the identity, so the only element fixing all roots is the identity. ^{[Bourbaki Ch VI; Humphreys §8-10; Serre]}

Bridge. The Weyl group constructed here acts on the same root vectors that 07.06.05 encodes as Dynkin diagrams, with each diagram node corresponding to a simple root and each edge recording the angle between them. The finiteness of $W$ is the structural fact that makes the classification in 07.04.01 terminate in finitely many families. This reflection symmetry also controls the weight lattice in 07.06.01, where weights related by a Weyl-group element have the same dimension and character.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

Mathlib contains related infrastructure, but the exact theorem package for this unit is only partially represented in the current Codex Lean layer.

import Mathlib

namespace Codex.RepresentationTheory.LieAlgebraic

theorem RootSystem_placeholder : True := by
  trivial

end Codex.RepresentationTheory.LieAlgebraic

Advanced results [Master]

The classification theorem for root systems states that every irreducible root system belongs to one of four infinite families $A_n$ ($n\ge 1$), $B_n$ ($n\ge 2$), $C_n$ ($n\ge 3$), $D_n$ ($n\ge 4$), or one of five exceptional types $G_2$, $F_4$, $E_6$, $E_7$, $E_8$. The families correspond to the classical Lie algebras: $A_n$ to $\mathfrak{sl}(n+1)$, $B_n$ to $\mathfrak{so}(2n+1)$, $C_n$ to $\mathfrak{sp}(2n)$, $D_n$ to $\mathfrak{so}(2n)$.

A Dynkin diagram encodes the simple roots and their pairwise angles. Nodes represent simple roots. Two nodes are unconnected if the angle is $90$ degrees, connected by a single line if $120$ degrees, a double line if $135$ degrees (with an arrow toward the shorter root), and a triple line if $150$ degrees. The classification of root systems reduces to the classification of connected Dynkin diagrams, a finite combinatorial problem. ^{[Bourbaki Ch VI; Humphreys §8-10; Serre]}

Synthesis. Root systems sit at the center of the Lie-algebraic strand, connecting structural information to representation-theoretic data. The root system of $\mathfrak{g}$ determines the weight lattice that classifies representations in 07.06.01, the Weyl group built from root reflections governs character formulas in 07.06.07, the Dynkin diagram of 07.06.05 provides the graphical encoding of the root system, and the Cartan matrix of 07.06.19 records the integer pairings between simple roots. The classification theorem shows that the infinitesimal symmetries of space — Lie algebras — are organized by a finite list of geometric patterns, a remarkable contraction from continuous complexity to discrete taxonomy.

Full proof set [Master]

The finiteness of the Weyl group was proved in the Key theorem section. The full classification of root systems proceeds by reducing to Dynkin diagrams. The key step is that every root system decomposes uniquely as an orthogonal direct sum of irreducible root systems, and each irreducible system has a connected Dynkin diagram. The proof that the list $A_n,B_n,C_n,D_n,E_6,E_7,E_8,F_4,G_2$ is exhaustive uses the crystallographic restriction (R3) to bound the possible angles between simple roots, and then a combinatorial argument on weighted graphs eliminates all configurations except the known diagrams. Full details appear in Humphreys Chapters 10–12 and Bourbaki Chapter VI. ^{[Bourbaki Ch VI; Humphreys §8-10; Serre]}

Connections [Master]

• 07.06.01 uses root systems to define weight lattices and highest weights; the roots are the nonzero weights of the adjoint representation.
• 07.06.05 encodes each root system as a Dynkin diagram, the graphical signature of the Lie algebra.
• 07.06.04 constructs the Weyl group from root reflections and develops its combinatorial structure.
• 07.04.01 uses root system data in the Cartan–Weyl classification of semisimple Lie algebras.
• 03.04.01 introduces the Lie algebra whose adjoint representation produces the root decomposition.

Bibliography [Master]

• Killing, W. Die Zusammensetzung der stetigen endlichen Transformationsgruppen, Mathematische Annalen, 1888–1890.
• Cartan 1894 Sur la structure des groupes de transformations finis et continus.
• Bourbaki, N. Lie Groups and Lie Algebras, Chapters 4–6, Springer.
• Humphreys, J.E. Introduction to Lie Algebras and Representation Theory, Springer GTM 9.
• Serre, J.-P. Complex Semisimple Lie Algebras, Springer.

Historical & philosophical context [Master]

Wilhelm Killing discovered root systems between 1888 and 1890 while attempting to classify all finite-dimensional Lie algebras over the complex numbers. His calculations contained gaps and errors, but the structural picture — a Cartan subalgebra, root spaces, and the integrality conditions — was correct. Cartan revised and completed Killing's classification in his 1894 thesis, establishing the root system as the central structural invariant of a semisimple Lie algebra. The axiomatic definition of a root system as an independent geometric object is due to Bourbaki in the 1940s and 1950s, who separated root-system theory from its Lie-algebraic origins. This abstraction revealed that root systems appear in many other contexts: singularity theory, cluster algebras, and combinatorial representation theory. ^{[Killing 1888-90; Cartan 1894; Bourbaki Ch VI]}