An interface for Lie algebras

LieAlgebra{𝔽, G} <: AbstractManifold{𝔽}

Represent the Lie algebra $\mathfrak g$, that is a $𝔽$ vector space with an associated lie_bracket $[⋅,⋅]: \mathfrak g×\mathfrak g → \mathfrak g$ which fulfills

1. $[X,X] = 0$ for all $X ∈ \mathfrak g$
2. The Jacobi identity $[X, [Y,Z]] = [[X,Y],Z] = [Y, [X,Z]]$ holds for all $X, Y, Z ∈ \mathfrak g$.

The Lie algebras considered here are those related to a LieGroup $\mathcal G$, namely the tangent space $T_{\mathrm{e}}\mathcal G$ at the Identity, this is internally just a const of the corresponding TangentSpace.

Constructor

LieAlgebra(G::LieGroup)

Return the Lie Algebra belonging to the LieGroup G.

LieAlgebraOrthogonalBasis{𝔽} <: ManifoldsBase.AbstractOrthogonalBasis{𝔽,ManifoldsBase.TangentSpaceType}

Specify an orthogonal basis for a Lie algebra. This is used as the default within hat and vee.

If not specifically overwritten/implemented for a Lie group, the DefaultOrthogonalBasis at the identity_element on the `base_manifold acts as a fallback.

rand(::LieGroup; vector_at=nothing, σ=1.0, kwargs...)
rand(::LieAlgebra; σ=1.0, kwargs...)
rand!(::LieGroup, gX; vector_at=nothing, kwargs...)
rand!(::LieAlgebra, X; σ=1.0, kwargs...)

Compute a random point or tangent vector on a Lie group.

For points this just means to generate a random point on the underlying manifold itself.

For tangent vectors, an element in the Lie Algebra is generated, see also rand(::LieAlgebra; kwargs...)

lie_bracket!(𝔤::LieAlgebra, X, Y)
lie_bracket!(𝔤::LieAlgebra, Z, X, Y)

Compute the Lie bracket $[⋅,⋅]: \mathfrak g×\mathfrak g → \mathfrak g$ which fulfills

1. $[X,X] = 0$ for all $X ∈ \mathfrak g$
2. The Jacobi identity $[X, [Y,Z]] = [[X,Y],Z] = [Y, [X,Z]]$ holds for all $X, Y, Z ∈ \mathfrak g$.

The computation can be done in-place of Z.

lie_bracket!(𝔤::LieAlgebra, X, Y)
lie_bracket!(𝔤::LieAlgebra, Z, X, Y)

Compute the Lie bracket $[⋅,⋅]: \mathfrak g×\mathfrak g → \mathfrak g$ which fulfills

1. $[X,X] = 0$ for all $X ∈ \mathfrak g$
2. The Jacobi identity $[X, [Y,Z]] = [[X,Y],Z] = [Y, [X,Z]]$ holds for all $X, Y, Z ∈ \mathfrak g$.

The computation can be done in-place of Z.

is_point(𝔤::LieAlgebra, X; kwargs...)

Check whether X is a valid point on the Lie Algebra 𝔤. This falls back to checking whether X is a valid point on the tangent space at the identity_element(G) on G.manifold on the LieGroup of G

rand(::LieGroup; vector_at=nothing, σ=1.0, kwargs...)
rand(::LieAlgebra; σ=1.0, kwargs...)
rand!(::LieGroup, gX; vector_at=nothing, kwargs...)
rand!(::LieAlgebra, X; σ=1.0, kwargs...)

Compute a random point or tangent vector on a Lie group.

For points this just means to generate a random point on the underlying manifold itself.

For tangent vectors, an element in the Lie Algebra is generated, see also rand(::LieAlgebra; kwargs...)