Group actions

AbstractGroupAction{T<:GroupType, F}

An abstract type representing a group action. It is used to define how a group acts on a given vector space.

group(::AbstractGroupAction) -> AbstractGroup

Returns the group associated with a given AbstractGroupAction.

algebra(a::AbstractGroupAction{Lie}) -> AbstractLieAlgebra

Returns the associated Lie algebra of the Lie group of the action.

MatrixGroupAction{T<:GroupType, F} <: AbstractGroupAction{T, F}

Represents a group action of a matrix group on a set of variables.

Constructors

MatrixGroupAction(G::S, vectors::AbstractVector{<:AbstractVector{V}}) where {S<:AbstractGroup, V<:Variable}

Examples

julia> @polyvar x[1:3] y[1:3];

julia> SO3 = LieGroup("SO", 3);

julia> MatrixGroupAction(SO3, [x, y])
MatrixGroupAction of SO(3)
 2 vectors under action: [x₁, x₂, x₃], [y₁, y₂, y₃]

ScalingLieGroupAction <: AbstractGroupAction

Represents an action of a scaling Lie group on a set of variables.

Constructors

ScalingLieGroupAction(v::Vector{<:Variable})
ScalingLieGroupAction(V::AbstractMatrix{<:Variable})
ScalingLieGroupAction(G::ScalingLieGroup, v::Vector{<:Variable})

Examples

julia> @polyvar x[1:2, 1:3];

julia> ScalingLieGroupAction(x)
ScalingLieGroupAction of (ℂˣ)³
 vector under action: [x₁₋₁, x₂₋₁, x₁₋₂, x₂₋₂, x₁₋₃, x₂₋₃]
 action:
  x₁₋₁ ↦ λ₁x₁₋₁, x₂₋₁ ↦ λ₁x₂₋₁
  x₁₋₂ ↦ λ₂x₁₋₂, x₂₋₂ ↦ λ₂x₂₋₂
  x₁₋₃ ↦ λ₃x₁₋₃, x₂₋₃ ↦ λ₃x₂₋₃

julia> ScalingLieGroupAction(x[:])
ScalingLieGroupAction of ℂˣ
 vector under action: [x₁₋₁, x₂₋₁, x₁₋₂, x₂₋₂, x₁₋₃, x₂₋₃]
 action:
  x₁₋₁ ↦ λx₁₋₁, x₂₋₁ ↦ λx₂₋₁, x₁₋₂ ↦ λx₁₋₂, x₂₋₂ ↦ λx₂₋₂, x₁₋₃ ↦ λx₁₋₃, x₂₋₃ ↦ λx₂₋₃

DirectProductGroupAction <: AbstractGroupAction

Represents an action of a direct product group on a vector space.

Examples

julia> @polyvar x[1:3, 1:2];

julia> SO3 = LieGroup("SO", 3);

julia> a₁ = MatrixGroupAction(SO3, eachcol(x))
MatrixGroupAction of SO(3)
 2 vectors under action: [x₁₋₁, x₂₋₁, x₃₋₁], [x₁₋₂, x₂₋₂, x₃₋₂]

julia> a₂ = ScalingLieGroupAction(x)
ScalingLieGroupAction of (ℂˣ)²
 vector under action: [x₁₋₁, x₂₋₁, x₃₋₁, x₁₋₂, x₂₋₂, x₃₋₂]
 action:
  x₁₋₁ ↦ λ₁x₁₋₁, x₂₋₁ ↦ λ₁x₂₋₁, x₃₋₁ ↦ λ₁x₃₋₁
  x₁₋₂ ↦ λ₂x₁₋₂, x₂₋₂ ↦ λ₂x₂₋₂, x₃₋₂ ↦ λ₂x₃₋₂

julia> a₁ × a₂
DirectProductGroupAction of SO(3) × (ℂˣ)²
 lie actions: