# Lie algebra homomorphisms

Homomorphisms of Lie algebras in Oscar are represented by the type LieAlgebraHom.

## Constructors

Lie algebra homomorphisms $h: L_1 \to L_2$ are constructed by providing either the images of the basis elements of $L_1$ or a $\dim L_1 \times \dim L_2$ matrix.

homMethod
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)
source
homMethod
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)
source
identity_mapMethod
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
source

## Functions

The following functions are available for LieAlgebraHoms:

### Basic properties

For a homomorphism $h: L_1 \to L_2$, domain(h) and codomain(h) return $L_1$ and $L_2$ respectively.

matrixMethod
matrix(h::LieAlgebraHom) -> MatElem

Return the transformation matrix of h w.r.t. the bases of the domain and codomain.

Note: The matrix operates on the coefficient vectors from the right.

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### Image

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

Return the image of x under h.

source
imageMethod
image(h::LieAlgebraHom) -> LieSubalgebra

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

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

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

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

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

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### Kernel

kernelMethod
kernel(h::LieAlgebraHom) -> LieAlgebraIdeal

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

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### Composition

composeMethod
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.

source

### Inverses

is_isomorphismMethod
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.

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invMethod
inv(h::LieAlgebraHom) -> LieAlgebraHom

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

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