Lie.jl

A Julia package for computations with semisimple Lie algebras: root systems, Weyl groups, weight lattices, and representation-theoretic operations.

Features

Module	Highlights
DynkinTypes	Type-level classification TypeA{N}, …, TypeG2, ProductDynkinType
CartanMatrix	Compile-time @generated Cartan matrices, symmetrizers, bilinear forms
RootSystem	Positive roots, coroots, reflection tables (immutable singletons)
WeightLattice	Fundamental weights, Weyl vector, dominance, conjugation to dominant chamber
WeylGroup	Reduced words, multiplication via reflection tables, orbits, Weyl dimension formula, Borel–Weil–Bott
Characters	WeylCharacter (representation ring), Freudenthal formula, Brauer–Klimyk tensor products, Adams operators, symmetric/exterior powers

Installation

using Pkg
Pkg.add(url="https://github.com/HomogeneousTools/Lie.jl")  # adjust URL

Or for local development:

julia --project=.

Quick start

using Lie

# Fundamental weights of A₃
ω₁ = fundamental_weight(TypeA{3}, 1)
ω₂ = fundamental_weight(TypeA{3}, 2)

# Dimension of the standard representation
degree(ω₁)   # 4

# Tensor product decomposition: V(ω₁) ⊗ V(ω₁) = Sym²V ⊕ ⋀²V
tensor_product(ω₁, ω₁)

# Symmetric and exterior powers
Sym(3, ω₁)    # Sym³(standard rep)
⋀(2, ω₁)      # ⋀²(standard rep) = V(ω₂)

# Weyl orbit and dimension
length(weyl_orbit(TypeA{3}, ω₁))   # 4
weyl_order(TypeA{3})                # 24

# Borel–Weil–Bott theorem
borel_weil_bott(ω₁)   # (0, ω₁) — degree 0, weight ω₁

# Virtual (Weyl) characters and the representation ring
V = WeylCharacter(ω₁)
V * V                    # tensor product as ring multiplication
dual(V)                  # contragredient
is_effective(V - V)      # false (zero character, but not a "negative" rep)

Running tests

julia --project=. -e 'using Pkg; Pkg.test()'

Running doctests

Doctests are embedded in docstrings throughout the source. Run them with:

julia --project=. -e 'using Documenter, Lie; doctest(Lie)'

Or via the docs build:

julia --project=docs/ docs/make.jl

Running benchmarks

# Run benchmarks and save results to benchmark/results/<timestamp>.json
julia --project=. benchmark/bench.jl

# Run and compare against the previous saved baseline
julia --project=. benchmark/bench.jl --compare

Benchmark results track minimum time, allocation count, and memory usage across 9 categories (w₀ action, conjugation, Weyl orbits, dimension formula, Freudenthal, tensor products, exterior/symmetric powers, Borel–Weil–Bott, infrastructure). Results are saved as JSON for regression tracking.

Dependencies

• StaticArrays.jl — fixed-size arrays for compile-time root/weight data
• LinearAlgebra (stdlib)
• BenchmarkTools.jl — benchmarking
• Documenter.jl — documentation and doctests

Julia ≥ 1.9 required.

License

See LICENSE for details.