LieAlgebras: Add zero_map

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

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

experimental/LieAlgebras/src/LieAlgebraHom.jl

@@ -301,3 +301,30 @@ 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

experimental/LieAlgebras/src/LieAlgebraModuleHom.jl

@@ -298,6 +298,35 @@ function identity_map(V::LieAlgebraModule)
   return hom(V, V, basis(V); check=false)
 end
 
+@doc raw"""
+    zero_map(V1::LieAlgebraModule, V2::LieAlgebraModule) -> LieAlgebraModuleHom
+    zero_map(V::LieAlgebraModule) -> LieAlgebraModuleHom
+
+Construct the zero map from `V1` to `V2` or from `V` to `V`.
+
+# Examples
+```jldoctest
+julia> L = special_linear_lie_algebra(QQ, 3);
+
+julia> V = standard_module(L)
+Standard module
+  of dimension 3
+over special linear Lie algebra of degree 3 over QQ
+
+julia> zero_map(V)
+Lie algebra module morphism
+  from standard module of dimension 3 over sl_3
+  to   standard module of dimension 3 over sl_3
+```
+"""
+function zero_map(V1::LieAlgebraModule{C}, V2::LieAlgebraModule{C}) where {C<:RingElement}
+  return hom(V1, V2, zero_matrix(coefficient_ring(V2), dim(V1), dim(V2)); check=false)
+end
+
+function zero_map(V::LieAlgebraModule)
+  return zero_map(V, V)
+end
+
 ###############################################################################
 #
 #   Hom constructions

experimental/LieAlgebras/src/LieAlgebras.jl

@@ -57,6 +57,7 @@ import ..Oscar:
   symbols,
   symmetric_power,
   tensor_product,
+  zero_map,
   ⊕,
   ⊗
 

experimental/LieAlgebras/test/LieAlgebraHom-test.jl

@@ -37,6 +37,31 @@
     @test h(ideal(L, [x2, x3, x4])) == sub(L, [x3])
   end
 
+  @testset "Identity and zero map" begin
+    L1 = special_linear_lie_algebra(QQ, 3)
+    L2 = general_linear_lie_algebra(QQ, 3)
+
+    h = identity_map(L1)
+    @test domain(h) == L1
+    @test codomain(h) == L1
+    @test matrix(h) == identity_matrix(QQ, dim(L1))
+    for x in basis(L1)
+      @test h(x) == x
+    end
+    @test image(h) == sub(L1)
+    @test kernel(h) == ideal(L1, [zero(L1)])
+
+    h = zero_map(L1, L2)
+    @test domain(h) == L1
+    @test codomain(h) == L2
+    @test matrix(h) == zero_matrix(QQ, dim(L1), dim(L2))
+    for x in basis(L1)
+      @test h(x) == zero(L2)
+    end
+    @test image(h) == sub(L2, [zero(L2)])
+    @test kernel(h) == ideal(L1)
+  end
+
   @testset "Composition" begin
     L1 = special_linear_lie_algebra(QQ, 2)
     L2 = special_linear_lie_algebra(QQ, 3)
@@ -76,6 +101,8 @@
 
     @test is_isomorphism(identity_map(L1))
     @test identity_map(L1) == inv(identity_map(L1))
+
+    @test !is_isomorphism(zero_map(L1))
   end
 
   @testset "Hum72, Exercise 2.10" begin