𝓜

Math Notes

University Mathematics

ConceptComplete

Weight Lattices and Root Systems

Root systems provide the combinatorial foundation for classification of semisimple Lie algebras and their representations.

DefinitionRoot System

A root system in a Euclidean space EE is a finite set Φ⊂E\Phi \subset E of non-zero vectors (roots) satisfying:

  1. Φ\Phi spans EE
  2. For α∈Φ\alpha \in \Phi, the only scalar multiples in Φ\Phi are ±α\pm\alpha
  3. For α∈Φ\alpha \in \Phi, the reflection sαs_\alpha preserves Φ\Phi, where sα(β)=β−2⟨β,α⟩⟨α,α⟩αs_\alpha(\beta) = \beta - 2\frac{\langle \beta, \alpha \rangle}{\langle \alpha, \alpha \rangle} \alpha
  4. For α,β∈Φ\alpha, \beta \in \Phi, we have 2⟨β,α⟩⟨α,α⟩∈Z2\frac{\langle \beta, \alpha \rangle}{\langle \alpha, \alpha \rangle} \in \mathbb{Z}
ExampleRoot Systems of Classical Lie Algebras

Type AnA_n (sln+1\mathfrak{sl}_{n+1}): Roots are ei−eje_i - e_j for i≠ji \neq j, where {e1,…,en+1}\{e_1, \ldots, e_{n+1}\} is the standard basis with ∑ei=0\sum e_i = 0.

Type BnB_n (so2n+1\mathfrak{so}_{2n+1}): Roots are ±ei±ej\pm e_i \pm e_j (i≠ji \neq j) and ±ei\pm e_i.

Type CnC_n (sp2n\mathfrak{sp}_{2n}): Roots are ±ei±ej\pm e_i \pm e_j (i≠ji \neq j) and ±2ei\pm 2e_i.

Type DnD_n (so2n\mathfrak{so}_{2n}): Roots are ±ei±ej\pm e_i \pm e_j for i≠ji \neq j.

DefinitionSimple Roots and Fundamental Weights

Choose a set of positive roots Φ+\Phi^+. The simple roots {α1,…,αn}\{\alpha_1, \ldots, \alpha_n\} are the indecomposable positive roots (not sums of other positive roots).

The fundamental weights {ω1,…,ωn}\{\omega_1, \ldots, \omega_n\} satisfy: 2⟨ωi,αj⟩⟨αj,αj⟩=δij2\frac{\langle \omega_i, \alpha_j \rangle}{\langle \alpha_j, \alpha_j \rangle} = \delta_{ij}

Dominant integral weights are non-negative integer combinations of fundamental weights: λ=∑i=1nmiωi\lambda = \sum_{i=1}^n m_i \omega_i with mi≥0m_i \geq 0.

TheoremWeyl Group

The Weyl group WW is generated by reflections sαs_\alpha for α∈Φ\alpha \in \Phi. It acts on the weight lattice and satisfies:

  1. WW is a finite Coxeter group
  2. WW preserves the root system: w(Φ)=Φw(\Phi) = \Phi for all w∈Ww \in W
  3. Irreducible characters are invariant: χL(λ)(w⋅g)=χL(λ)(g)\chi_{L(\lambda)}(w \cdot g) = \chi_{L(\lambda)}(g) (conjugacy invariance)

The Weyl group for AnA_n is Sn+1S_{n+1}, for Bn,CnB_n, C_n is the hyperoctahedral group, and for DnD_n is a subgroup of the hyperoctahedral group.

Remark

Root systems classify semisimple Lie algebras:

  • Dynkin diagrams: Encode the Cartan matrix and determine the Lie algebra up to isomorphism
  • Classification: Four infinite families (An,Bn,Cn,Dn)(A_n, B_n, C_n, D_n) and five exceptionals (E6,E7,E8,F4,G2)(E_6, E_7, E_8, F_4, G_2)
  • Geometry: Root systems appear in crystallography, sphere packing, and lattice theory
  • Physics: Gauge groups in particle physics correspond to Lie algebras with specific root systems

The combinatorics of root systems, Weyl groups, and weight lattices provide computational tools for representation theory while revealing deep structural patterns.