LieAlgebras: add dual modules

experimental/LieAlgebras/src/LieAlgebraModule.jl

@@ -126,6 +126,11 @@ function show_standard_module(io::IO, V::LieAlgebraModule{C}) where {C<:RingElem
   print(IOContext(io, :compact => true), base_lie_algebra(V))
 end
 
+function show_dual(io::IO, V::LieAlgebraModule{C}) where {C<:RingElement}
+  print(io, "Dual of ")
+  print(IOContext(io, :compact => true), get_attribute(V, :base_module))
+end
+
 function show_exterior_power(io::IO, V::LieAlgebraModule{C}) where {C<:RingElement}
   print(io, "$(get_attribute(V, :power))-th exterior power of ")
   print(IOContext(io, :compact => true), base_module(V))
@@ -189,6 +194,9 @@ function (V::LieAlgebraModule{C})(v::SRow{C}) where {C<:RingElement}
 end
 
 function (V::LieAlgebraModule{C})(v::LieAlgebraModuleElem{C}) where {C<:RingElement}
+  if is_dual(V) && base_module(V) == parent(v)
+    return V(coefficients(v))
+  end
   @req V == parent(v) "Incompatible modules."
   return v
 end
@@ -336,6 +344,10 @@ function is_standard_module(V::LieAlgebraModule{C}) where {C<:RingElement}
   return get_attribute(V, :type, :fallback) == :standard_module
 end
 
+function is_dual(V::LieAlgebraModule{C}) where {C<:RingElement}
+  return get_attribute(V, :type, :fallback) == :dual
+end
+
 function is_exterior_power(V::LieAlgebraModule{C}) where {C<:RingElement}
   return get_attribute(V, :type, :fallback) == :exterior_power
 end
@@ -349,7 +361,7 @@ function is_tensor_power(V::LieAlgebraModule{C}) where {C<:RingElement}
 end
 
 function base_module(V::LieAlgebraModule{C}) where {C<:RingElement}
-  @req is_exterior_power(V) || is_symmetric_power(V) || is_tensor_power(V) "Not a power module."
+  @req is_dual(V) || is_exterior_power(V) || is_symmetric_power(V) || is_tensor_power(V) "Not a power module."
   return get_attribute(V, :base_module)
 end
 
@@ -407,6 +419,26 @@ function standard_module(L::LinearLieAlgebra{C}) where {C<:RingElement}
   return std_V
 end
 
+function dual(V::LieAlgebraModule{C}) where {C<:RingElement}
+  L = base_lie_algebra(V)
+  dim_dual_V = dim(V)
+
+  transformation_matrices = map(1:dim(L)) do i
+    -transpose(transformation_matrix(V, i))
+  end
+
+  if is_standard_module(V)
+    parentheses = identity
+  else
+    parentheses = x -> "($x)"
+  end
+  s = [Symbol("$(parentheses(s))*") for s in symbols(V)]
+
+  pow_V = LieAlgebraModule{C}(L, dim_dual_V, transformation_matrices, s; check=false)
+  set_attribute!(pow_V, :type => :dual, :base_module => V, :show => show_dual)
+  return pow_V
+end
+
 function exterior_power(V::LieAlgebraModule{C}, k::Int) where {C<:RingElement}
   L = base_lie_algebra(V)
   dim_pow_V = binomial(dim(V), k)

experimental/LieAlgebras/src/LieAlgebras.jl

@@ -17,6 +17,7 @@ import ..Oscar:
   coeff,
   coefficients,
   dim,
+  dual,
   elem_type,
   expressify,
   exterior_power,
@@ -43,6 +44,7 @@ export combinations
 export exterior_power
 export general_linear_lie_algebra
 export highest_weight_module
+export is_dual
 export is_exterior_power
 export is_standard_module
 export is_symmetric_power

experimental/LieAlgebras/test/LieAlgebraModule-test.jl

@@ -22,6 +22,8 @@ function lie_algebra_module_conformance_test(
     @test base_ring(v) == base_ring(V)
     @test elem_type(base_ring(V)) == C
 
+    @test base_lie_algebra(V) === L
+
     # this block stays only as long as `ngens` and `gens` are not specialized for Lie algebra modules
     @test dim(V) == ngens(V)
     @test basis(V) == gens(V)
@@ -202,6 +204,7 @@ end
     @test length(repr(V)) < 10^4 # outputs tend to be excessively long due to recursion
 
     @test is_standard_module(V)
+    @test !is_dual(V)
     @test !is_exterior_power(V)
     @test !is_symmetric_power(V)
     @test !is_tensor_power(V)
@@ -211,6 +214,33 @@ end
     @test x * v == V(matrix_repr(x) * coefficients(v))
   end
 
+  @testset "dual" begin
+    L = special_orthogonal_lie_algebra(QQ, 4)
+    V = symmetric_power(standard_module(L), 2)
+    type_V = module_type_bools(V)
+
+    dual_V = dual(V)
+    @test type_V == module_type_bools(V) # construction of dual_V should not change type of V
+    @test base_module(dual_V) === V
+    @test dim(dual_V) == dim(V)
+    @test length(repr(dual_V)) < 10^4 # outputs tend to be excessively long due to recursion
+
+    @test !is_standard_module(dual_V)
+    @test is_dual(dual_V)
+    @test !is_exterior_power(dual_V)
+    @test !is_symmetric_power(dual_V)
+    @test !is_tensor_power(dual_V)
+
+    dual_dual_V = dual(dual_V)
+    @test dim(dual_dual_V) == dim(V)
+    @test all(
+      i ->
+        Oscar.LieAlgebras.transformation_matrix(dual_dual_V, i) ==
+        Oscar.LieAlgebras.transformation_matrix(V, i),
+      1:dim(L),
+    )
+  end
+
   @testset "exterior_power" begin
     L = special_orthogonal_lie_algebra(QQ, 4)
     V = symmetric_power(standard_module(L), 2)
@@ -224,6 +254,7 @@ end
       @test length(repr(pow_V)) < 10^4 # outputs tend to be excessively long due to recursion
 
       @test !is_standard_module(pow_V)
+      @test !is_dual(V)
       @test is_exterior_power(pow_V)
       @test !is_symmetric_power(pow_V)
       @test !is_tensor_power(pow_V)
@@ -256,6 +287,7 @@ end
       @test length(repr(pow_V)) < 10^4 # outputs tend to be excessively long due to recursion
 
       @test !is_standard_module(pow_V)
+      @test !is_dual(V)
       @test !is_exterior_power(pow_V)
       @test is_symmetric_power(pow_V)
       @test !is_tensor_power(pow_V)
@@ -288,6 +320,7 @@ end
       @test length(repr(pow_V)) < 10^4 # outputs tend to be excessively long due to recursion
 
       @test !is_standard_module(pow_V)
+      @test !is_dual(V)
       @test !is_exterior_power(pow_V)
       @test !is_symmetric_power(pow_V)
       @test is_tensor_power(pow_V)