Lie algebra homomorphisms

hom(L1::LieAlgebra, L2::LieAlgebra, imgs::Vector{<:LieAlgebraElem}; check::Bool=true) -> LieAlgebraHom

Construct the homomorphism from L1 to L2 by sending the i-th basis element of L1 to imgs[i] and extending linearly. All elements of imgs must lie in L2.

By setting check=false, the linear map is not checked to be compatible with the Lie bracket.

Examples

julia> L1 = special_linear_lie_algebra(QQ, 2);

julia> L2 = special_linear_lie_algebra(QQ, 3);

julia> h = hom(L1, L2, [basis(L2, 1), basis(L2, 4), basis(L2, 7)]) # embed sl_2 into sl_3
Lie algebra morphism
  from special linear Lie algebra of degree 2 over QQ
  to special linear Lie algebra of degree 3 over QQ
  
julia> [(x, h(x)) for x in basis(L1)]
3-element Vector{Tuple{LinearLieAlgebraElem{QQFieldElem}, LinearLieAlgebraElem{QQFieldElem}}}:
 (e_1_2, e_1_2)
 (f_1_2, f_1_2)
 (h_1, h_1)

hom(L1::LieAlgebra, L2::LieAlgebra, mat::MatElem; check::Bool=true) -> LieAlgebraHom

Construct the homomorphism from L1 to L2 by acting with the matrix mat from the right on the coefficient vector w.r.t. the basis of L1. mat must be a matrix of size dim(L1) \times dim(L2) over coefficient_ring(L2).

By setting check=false, the linear map is not checked to be compatible with the Lie bracket.

Examples

julia> L1 = special_linear_lie_algebra(QQ, 2);

julia> L2 = general_linear_lie_algebra(QQ, 2);

julia> h = hom(L1, L2, matrix(QQ, [0 1 0 0; 0 0 1 0; 1 0 0 -1]))
Lie algebra morphism
  from special linear Lie algebra of degree 2 over QQ
  to general linear Lie algebra of degree 2 over QQ

julia> [(x, h(x)) for x in basis(L1)]
3-element Vector{Tuple{LinearLieAlgebraElem{QQFieldElem}, LinearLieAlgebraElem{QQFieldElem}}}:
 (e_1_2, x_1_2)
 (f_1_2, x_2_1)
 (h_1, x_1_1 - x_2_2)

identity_map(L::LieAlgebra) -> LieAlgebraHom

Construct the identity map on L.

Examples

julia> L = special_linear_lie_algebra(QQ, 3)
Special linear Lie algebra of degree 3
  of dimension 8
over rational field

julia> identity_map(L)
Lie algebra morphism
  from special linear Lie algebra of degree 3 over QQ
  to special linear Lie algebra of degree 3 over QQ

zero_map(L1::LieAlgebra, L2::LieAlgebra) -> LieAlgebraHom
zero_map(L::LieAlgebra) -> LieAlgebraHom

Construct the zero map from L1 to L2 or from L to L.

Examples

julia> L = special_linear_lie_algebra(QQ, 3)
Special linear Lie algebra of degree 3
  of dimension 8
over rational field

julia> zero_map(L)
Lie algebra morphism
  from special linear Lie algebra of degree 3 over QQ
  to special linear Lie algebra of degree 3 over QQ

matrix(R::Ring, arr::AbstractMatrix{T}) where {T}

Constructs the matrix over $R$ with entries as in arr.

Examples

julia> matrix(GF(3), [1 2 ; 3 4])
[1   2]
[0   1]

julia> using LinearAlgebra ; matrix(GF(5), I(2))
[1   0]
[0   1]

matrix(R::Ring, r::Int, c::Int, arr::AbstractVector{T}) where {T}

Constructs the $r \times c$ matrix over $R$, where the entries are taken row-wise from arr.

image(h::LieAlgebraHom, x::LieAlgebraElem) -> LieAlgebraElem

Return the image of x under h.

image(h::LieAlgebraHom) -> LieSubalgebra

Return the image of h as a Lie subalgebra of the codomain.

image(h::LieAlgebraHom, I::LieAlgebraIdeal) -> LieSubalgebra

Return the image of I under h as a Lie subalgebra of the codomain.

image(h::LieAlgebraHom, S::LieSubalgebra) -> LieSubalgebra

Return the image of S under h as a Lie subalgebra of the codomain.

kernel(h::LieAlgebraHom) -> LieAlgebraIdeal

Return the kernel of h as an ideal of the domain.

compose(f::LieAlgebraHom, g::LieAlgebraHom) -> LieAlgebraHom

Return the composition of f and g, i.e. the homomorphism h such that h(x) = g(f(x)) for all x in the domain of f. The codomain of f must be identical to the domain of g.

is_isomorphism(h::LieAlgebraHom) -> Bool

Return true if h is an isomorphism. This function tries to invert the transformation matrix of h and caches the result. The inverse isomorphism can be cheaply accessed via inv(h) after calling this function.

inv(h::LieAlgebraHom) -> LieAlgebraHom

Return the inverse of h. Requires h to be an isomorphism.