Module root_systems 

Root Systems for Semisimple Lie Algebras

Implements the theory of root systems, which classify semisimple Lie algebras via Cartan-Killing classification. Root systems encode the structure of the Lie bracket via the adjoint representation on the Cartan subalgebra.

Mathematical Background

For a semisimple Lie algebra 𝔤, choose a Cartan subalgebra 𝔥 (maximal abelian). The adjoint representation ad: 𝔤 → End(𝔤) diagonalizes on 𝔥, giving eigenvalues called roots:

[H, X_α] = α(H) X_α    for all H ∈ 𝔥

The set of roots Φ ⊂ 𝔥* forms a root system, satisfying:

1. Φ is finite, spans 𝔥*, and 0 ∉ Φ
2. If α ∈ Φ, then -α ∈ Φ and no other scalar multiples
3. Φ is closed under Weyl reflections: s_α(β) = β - 2⟨β,α⟩/⟨α,α⟩ · α
4. ⟨β,α⟩ ∈ ℤ for all α,β ∈ Φ (Cartan integers)

Cartan Classification

Type	Rank	Group	Dimension	Example
Aₙ	n	SU(n+1)	n(n+2)	A₁ = SU(2)
Bₙ	n	SO(2n+1)	n(2n+1)	B₂ = SO(5)
Cₙ	n	Sp(2n)	n(2n+1)	C₂ = Sp(4)
Dₙ	n	SO(2n)	n(2n-1)	D₃ = SO(6)
E₆	6	E₆	78	Exceptional
E₇	7	E₇	133	Exceptional
E₈	8	E₈	248	Exceptional
F₄	4	F₄	52	Exceptional
G₂	2	G₂	14	Exceptional

References

• Hall, Lie Groups, Lie Algebras, and Representations, Chapter 7
• Humphreys, Introduction to Lie Algebras and Representation Theory, Ch. II
• Fulton & Harris, Representation Theory, Lecture 21

Structs

Alcove

Fundamental alcove for an affine root system.

CartanSubalgebra

Cartan subalgebra of a semisimple Lie algebra.

Root

A root in the dual space of a Cartan subalgebra.

RootSystem

A root system for a semisimple Lie algebra.

WeightLattice

Weight lattice for a root system.

WeylChamber

Weyl chamber for a root system.