module_homs.md

CurrentModule = Oscar
DocTestSetup = quote
  using Oscar
end

Lie algebra module homomorphisms

Homomorphisms of Lie algebra modules in Oscar are represented by the type LieAlgebraModuleHom.

Constructors

Homomorphisms of modules over the same Lie algebra $h: V_1 \to V_2$ are constructed by providing either the images of the basis elements of $V_1$ or a $\dim V_1 \times \dim V_2$ matrix.

hom(::LieAlgebraModule{C}, ::LieAlgebraModule{C}, ::Vector{<:LieAlgebraModuleElem{C}}; check::Bool=true) where {C<:FieldElem}
hom(::LieAlgebraModule{C}, ::LieAlgebraModule{C}, ::MatElem{C}; check::Bool=true) where {C<:FieldElem}
identity_map(::LieAlgebraModule)
zero_map(::LieAlgebraModule{C}, ::LieAlgebraModule{C}) where {C<:FieldElem}

Functions

The following functions are available for LieAlgebraModuleHoms:

Basic properties

For a homomorphism $h: V_1 \to V_2$, domain(h) and codomain(h) return $V_1$ and $V_2$ respectively.

matrix(::LieAlgebraModuleHom{<:LieAlgebraModule,<:LieAlgebraModule{C2}}) where {C2<:FieldElem}

Image

image(::LieAlgebraModuleHom, ::LieAlgebraModuleElem)

Composition

compose(::LieAlgebraModuleHom{T1,T2}, ::LieAlgebraModuleHom{T2,T3}) where {T1<:LieAlgebraModule,T2<:LieAlgebraModule,T3<:LieAlgebraModule}

Inverses

is_isomorphism(::LieAlgebraModuleHom)
inv(::LieAlgebraModuleHom)

Hom constructions

Lie algebra module homomorphisms support + and - if they have the same domain and codomain.

canonical_injections(::LieAlgebraModule)
canonical_injection(::LieAlgebraModule, ::Int)
canonical_projections(::LieAlgebraModule)
canonical_projection(::LieAlgebraModule, ::Int)
hom_direct_sum(::LieAlgebraModule{C}, ::LieAlgebraModule{C}, ::Matrix{<:LieAlgebraModuleHom}) where {C<:FieldElem}
hom_tensor(::LieAlgebraModule{C}, ::LieAlgebraModule{C}, ::Vector{<:LieAlgebraModuleHom}) where {C<:FieldElem}
hom(::LieAlgebraModule{C}, ::LieAlgebraModule{C}, ::LieAlgebraModuleHom) where {C<:FieldElem}