LieAlgebras: Add is_welldefined for Lie algebra homs

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

experimental/LieAlgebras/test/LieAlgebraHom-test.jl

@@ -7,11 +7,7 @@
     @test domain(h) == L1
     @test codomain(h) == L2
     @test matrix(h) == matrix(QQ, [0 1 0 0; 0 0 1 0; 1 0 0 -1])
-    for x1 in basis(L1)
-      for x2 in basis(L1)
-        @test h(x1 * x2) == h(x1) * h(x2)
-      end
-    end
+    @test is_welldefined(h)
   end
 
   @testset "Image and kernel" begin
@@ -45,6 +41,7 @@
     @test domain(h) == L1
     @test codomain(h) == L1
     @test matrix(h) == identity_matrix(QQ, dim(L1))
+    @test is_welldefined(h)
     for x in basis(L1)
       @test h(x) == x
     end
@@ -55,6 +52,7 @@
     @test domain(h) == L1
     @test codomain(h) == L2
     @test matrix(h) == zero_matrix(QQ, dim(L1), dim(L2))
+    @test is_welldefined(h)
     for x in basis(L1)
       @test h(x) == zero(L2)
     end

experimental/LieAlgebras/test/LieAlgebraModuleHom-test.jl

@@ -33,6 +33,35 @@
     @test image(h, v6) == h(v6) == v3
   end
 
+  @testset "Identity and zero map" begin
+    L = special_linear_lie_algebra(QQ, 3)
+    stdV = standard_module(L)
+    V1 = symmetric_power(stdV, 3)
+    V2 = exterior_power(stdV, 2)
+
+    h = identity_map(V1)
+    @test domain(h) == V1
+    @test codomain(h) == V1
+    @test matrix(h) == identity_matrix(QQ, dim(V1))
+    @test is_welldefined(h)
+    for v in basis(V1)
+      @test h(v) == v
+    end
+    # @test image(h) == ...
+    # @test kernel(h) == ...
+
+    h = zero_map(V1, V2)
+    @test domain(h) == V1
+    @test codomain(h) == V2
+    @test matrix(h) == zero_matrix(QQ, dim(V1), dim(V2))
+    @test is_welldefined(h)
+    for v in basis(V1)
+      @test h(v) == zero(V2)
+    end
+    # @test image(h) == ...
+    # @test kernel(h) == ...
+  end
+
   @testset "Composition" begin
     L = special_orthogonal_lie_algebra(QQ, 3)
     V1 = standard_module(L)