LieAlgebras: Constructions for module homs (#2753)

oscar-system · Sep 19, 2023 · b4b9f68 · b4b9f68
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diff --git a/experimental/LieAlgebras/docs/src/lie_algebra_homs.md b/experimental/LieAlgebras/docs/src/lie_algebra_homs.md
@@ -19,6 +19,7 @@ the images of the basis elements of $L_1$ or a $\dim L_1 \times \dim L_2$ matrix
 hom(::LieAlgebra{C}, ::LieAlgebra{C}, ::Vector{<:LieAlgebraElem{C}}; check::Bool=true) where {C<:RingElement}
 hom(::LieAlgebra{C}, ::LieAlgebra{C}, ::MatElem{C}; check::Bool=true) where {C<:RingElement}
 identity_map(::LieAlgebra)
+zero_map(::LieAlgebra{C}, ::LieAlgebra{C}) where {C<:RingElement}
 ```
 
 ## Functions

diff --git a/experimental/LieAlgebras/docs/src/module_homs.md b/experimental/LieAlgebras/docs/src/module_homs.md
@@ -20,6 +20,7 @@ or a $\dim V_1 \times \dim V_2$ matrix.
 hom(::LieAlgebraModule{C}, ::LieAlgebraModule{C}, ::Vector{<:LieAlgebraModuleElem{C}}; check::Bool=true) where {C<:RingElement}
 hom(::LieAlgebraModule{C}, ::LieAlgebraModule{C}, ::MatElem{C}; check::Bool=true) where {C<:RingElement}
 identity_map(::LieAlgebraModule)
+zero_map(::LieAlgebraModule{C}, ::LieAlgebraModule{C}) where {C<:RingElement}
 ```
 
 ## Functions
@@ -48,3 +49,16 @@ compose(::LieAlgebraModuleHom{T1,T2}, ::LieAlgebraModuleHom{T2,T3}) where {T1<:L
 is_isomorphism(::LieAlgebraModuleHom)
 inv(::LieAlgebraModuleHom)
 ```
+
+### Hom constructions
+Lie algebra module homomorphisms support `+` and `-` if they have the same domain and codomain.
+
+```@docs
+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<:RingElement}
+hom_tensor(::LieAlgebraModule{C}, ::LieAlgebraModule{C}, ::Vector{<:LieAlgebraModuleHom}) where {C<:RingElement}
+hom_power(::LieAlgebraModule{C}, ::LieAlgebraModule{C}, ::LieAlgebraModuleHom) where {C<:RingElement}
+```
diff --git a/experimental/LieAlgebras/src/LieAlgebraHom.jl b/experimental/LieAlgebras/src/LieAlgebraHom.jl
@@ -26,9 +26,7 @@
     h.matrix = mat::dense_matrix_type(coefficient_ring(L2))
     h.header = MapHeader(L1, L2)
     if check
-      for x1 in basis(L1), x2 in basis(L1)
-        @req h(x1) * h(x2) == h(x1 * x2) "Not a homomorphism"
-      end
+      @req is_welldefined(h) "Not a homomorphism"
     end
     return h
   end
@@ -51,6 +49,20 @@ function matrix(h::LieAlgebraHom{<:LieAlgebra,<:LieAlgebra{C2}}) where {C2<:Ring
   return (h.matrix)::dense_matrix_type(C2)
 end
 
+@doc raw"""
+    is_welldefined(h::LieAlgebraHom) -> Bool
+
+Return `true` if `h` is a well-defined homomorphism of Lie algebras.
+This function is used internally when calling `hom` with `check=true`.
+"""
+function is_welldefined(h::LieAlgebraHom)
+  L1 = domain(h)
+  for x1 in basis(L1), x2 in basis(L1)
+    h(x1) * h(x2) == h(x1 * x2) || return false
+  end
+  return true
+end
+
 ###############################################################################
 #
 #   String I/O
@@ -301,3 +313,30 @@ Lie algebra morphism
 function identity_map(L::LieAlgebra)
   return hom(L, L, basis(L); check=false)
 end
+
+@doc raw"""
+    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
+```jldoctest
+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
+```
+"""
+function zero_map(L1::LieAlgebra{C}, L2::LieAlgebra{C}) where {C<:RingElement}
+  return hom(L1, L2, zero_matrix(coefficient_ring(L2), dim(L1), dim(L2)); check=false)
+end
+
+function zero_map(L::LieAlgebra)
+  return zero_map(L, L)
+end