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Semisimple.jl icon Semisimple.jl

A Julia package for computations with finite-dimensional complex semisimple Lie algebras via their root data: root systems, Weyl groups, weight lattices, and highest-weight representation-theoretic operations. It is heavily optimized and uses Julia's type system to specialize many finite root-data computations.

Download the PDF manual.

Features

  • Dynkin types — Type-level classification (TypeA{N}, TypeB{N}, …, TypeG2, products) with text Dynkin diagrams
  • Cartan matrices — Compile-time @generated Cartan matrices, symmetrisers, bilinear forms
  • Root systems — Positive roots, coroots, reflection tables (immutable singletons)
  • Weight lattice — Fundamental weights, Weyl vector, dominance, conjugation
  • Weyl groups — Reduced words, multiplication via reflection tables, orbits, dimension formula
  • Characters — Weyl characters (representation ring), Freudenthal formula, Brauer–Klimyk tensor products, Littlewood–Richardson (Type A), Adams operators, symmetric/exterior powers

Installation

using Pkg
Pkg.add(url="https://github.com/HomogeneousTools/Semisimple.jl")

Quick start

julia> using Semisimple

julia> ω1 = fundamental_weight(TypeA{3}, 1)
ω1

julia> degree(ω1)   # dimension of the standard representation
4

julia> V = WeylCharacter(ω1);

julia> tensor_product(V, V)   # V(ω1) ⊗ V(ω1) = Sym²V ⊕ ⋀²V
A3(2, 0, 0) + A3(0, 1, 0)

julia> Sym(2, V) + ⋀(2, V) == V * V   # Newton identity
true

julia> length(weyl_orbit(TypeA{3}, ω1))
4

Contents

Semisimple.Semisimple — Module
Semisimple

Julia package for computations with semisimple Lie algebras over ℂ.

Provides root systems, Weyl groups, and weight-lattice arithmetic for all classical and exceptional Dynkin types (A, B, C, D, E₆, E₇, E₈, F₄, G₂) as well as their direct products.

See the online documentation for usage examples and mathematical background.

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