Merge branch 'Dev'

Project.toml

@@ -1,7 +1,7 @@
 name = "TensorCategories"
 uuid = "258c694e-7338-4d4d-b524-4851272a75cf"
 authors = ["Fabian Mäurer"]
-version = "0.2.4"
+version = "0.2.5"
 
 [deps]
 InteractiveUtils = "b77e0a4c-d291-57a0-90e8-8db25a27a240"

src/CategoryFramework/AbstractMethods.jl

@@ -201,9 +201,10 @@ function direct_sum_decomposition(X::Object, S = simples(parent(X)))
 end
 
 function central_primitive_idempotents(H::AbstractHomSpace)
-    @assert H.X == H.Y "Not an endomorphism algebra"
+    @assert domain(H) == codomain(H) "Not an endomorphism algebra"
 
     A = endomorphism_ring(H.X, basis(H))
+    one(A)
     A.issemisimple = true
     idems = central_primitive_idempotents(A)
     [sum(basis(H) .* coefficients(i)) for i ∈ idems]
@@ -304,6 +305,7 @@ function _decompose_by_simple_endomorphism_ring(X::Object, E = End(X))
     R = endomorphism_ring(X, E)
     CR,_ = center(R)
     dR = dim(R)
+
     if !is_square(div(dR,dim(CR))) || dR == 1 || is_commutative(R) 
         return (X,1)
     end
@@ -313,6 +315,24 @@ function _decompose_by_simple_endomorphism_ring(X::Object, E = End(X))
     n,_ = size(matrix(G[1]))
     mats = matrix.(G)
 
+    for f ∈ E
+        eig = eigenvalues(f)
+
+        length(eig) < 1  && continue
+
+        Z = collect(values(eig))[1]
+
+        length(eig) == 1 && dim(Z) == dim(X) && continue
+
+        EZ = End(Z)
+
+        k = sqrt(int_dim(E)) - sqrt(int_dim(EZ))
+
+        (Z2, k2) = _decompose_by_simple_endomorphism_ring(Z,EZ)
+
+        return (Z2, k*k2)
+    end     
+
     for i ∈ 1:n
         m = matrix(K,n,length(G), hcat([M[i,:] for M ∈ mats]...))
 

src/TensorCategories.jl

@@ -36,6 +36,7 @@ import Oscar: +, @alias, @attributes, AbstractSet, AcbField, StructureConstantAl
 using InteractiveUtils
 using Memoization
 using SparseArrays
+using Base.Threads
 
 
 export - 

src/TensorCategoryFramework/Center/Center.jl

@@ -862,12 +862,12 @@ function image(f::CenterMorphism)
 end
 
 
-function left_inverse(f::CenterMorphism)
-    X = domain(f)
-    Y = codomain(f)
-    l_inv = central_projection(Y,X,left_inverse(morphsm(f)))
-    return Morphism(Y,X,l_inv)
-end
+# function left_inverse(f::CenterMorphism)
+#     X = domain(f)
+#     Y = codomain(f)
+#     l_inv = central_projection(Y,X,left_inverse(morphism(f)))
+#     return Morphism(Y,X,l_inv)
+# end
 
 function quotient(Y::CenterObject, X::Object)
     # TODO: Compute quotient
@@ -1030,20 +1030,21 @@ function hom_by_adjunction(X::CenterObject, Y::CenterObject)
     X_Homs = [Hom(object(X),s) for s ∈ S]
     Y_Homs = [Hom(s,object(Y)) for s ∈ S]
 
-    candidates = [int_dim(H)*int_dim(H2) > 0 for (s,H,H2) ∈ zip(S,X_Homs,Y_Homs)]
+    candidates = [int_dim(H)*int_dim(H2) > 0 for (H,H2) ∈ zip(X_Homs,Y_Homs)]
 
     !any(candidates) && return HomSpace(X,Y, CenterMorphism[]) 
 
-    # Take smalles s for Hom(X,I(s)) -> Hom(I(s), Y)
-    X_Homs = X_Homs[candidates]
-    Y_Homs = Y_Homs[candidates]
+
+    # X_Homs = X_Homs[candidates]
+    # Y_Homs = Y_Homs[candidates]
 
 
     M = zero_matrix(base_ring(C),0,*(size(matrix(zero_morphism(X,Y)))...))
 
     mors = []
 
-    for (s, X_s, s_Y) ∈ zip(S[candidates], X_Homs, Y_Homs)
+    @threads for i ∈ findall(==(true), candidates)
+        s, X_s, s_Y = S[i], X_Homs[i], Y_Homs[i]
         Is = induction(s, parent_category = Z)
         B = induction_right_adjunction(X_s, X, Is)
         B2 = induction_adjunction(s_Y, Y, Is)
@@ -1147,7 +1148,7 @@ function smatrix(C::CenterCategory)
     n = length(simpls)
     K = base_ring(C)
     S = [zero_morphism(category(C)) for _ ∈ 1:n, _ ∈ 1:n]
-    for i ∈ 1:n
+    @threads for i ∈ 1:n
         for j ∈ i:n
             S[i,j] = S[j,i] = tr(half_braiding(simpls[i], object(simpls[j])) ∘ half_braiding(simpls[j], object(simpls[i])))
         end

src/TensorCategoryFramework/Center/Induction.jl

@@ -16,12 +16,12 @@ function induction(X::Object, simples::Vector = simples(parent(X)); parent_categ
 
     C = parent(X)
 
-    for i ∈ 1:length(simples)
+    @threads for i ∈ 1:length(simples)
         W = simples[i]
         γ[i] = zero_morphism(zero(parent(X)), direct_sum([W⊗((s⊗X)⊗dual(s)) for s ∈ simples])[1])
 
         for S ∈ simples
-            dom_i = ((S⊗X)⊗dual(S))⊗simples[i]
+            dom_i = ((S⊗X)⊗dual(S))⊗W
             γ_i_temp = zero_morphism(dom_i, zero(parent(X)))
             for  T ∈ simples
                 #@show S,T,W
@@ -74,7 +74,7 @@ function induction(X::Object, simples::Vector = simples(parent(X)); parent_categ
         # factor out such that the left and right tensor product coincide.
         r = direct_sum([horizontal_direct_sum([image(f⊗id(X)⊗id(dual(b)) - id(b)⊗id(X)⊗dual(f))[2] for f in End(b)]) for b in simples])
 
-        @show Z,r = cokernel(r)
+        Z,r = cokernel(r)
         ir = right_inverse(r)
 
         γ = [(id(b)⊗r) ∘ γᵢ ∘ (ir ⊗ id(b)) for (γᵢ,b) ∈ zip(γ, simples)]
@@ -124,11 +124,17 @@ function end_of_induction(X::Object, IX = induction(X))
     B = basis(Hom(X,object(IX)))
     simpls = simples(parent(X))
 
-    m = [(dim(xi))*compose(
-        inv(half_braiding(IX, xi)) ⊗ id(dual(xi)),
+    m = [zero_morphism(X,X) for _ in 1:length(simpls)]
+
+    @threads for i ∈ 1:length(simpls)
+        xi = simpls[i]
+        dxi = dual(xi)
+        m[i] = (dim(xi))*compose(
+        inv(half_braiding(IX, xi)) ⊗ id(dxi),
         associator(object(IX), xi, dual(xi)),
-        id(object(IX)) ⊗ (ev(dual(xi)) ∘ (spherical(xi) ⊗ id(dual(xi))))
-    ) for xi ∈ simpls]
+        id(object(IX)) ⊗ (ev(dxi) ∘ (spherical(xi) ⊗ id(dxi)))
+        ) 
+    end
 
     m = horizontal_direct_sum(m)
 
@@ -368,39 +374,62 @@ function adjusted_dual_basis(V::Vector{<:Morphism}, U::Vector{<:Morphism}, S::Ob
     return V, dual_basis
 end
 
-function partial_induction(X::Object, ind_simples::Vector, simples::Vector = simples(parent(X)); parent_category::CenterCategory = Center(parent(X)))
+function relative_induction(X::Object, ind_simples::Vector{<:Object}, simples::Vector = simples(parent(X)); parent_category::CenterCategory = Center(parent(X)))
+
+    if isdefined(parent_category, :inductions) && X ∈ collect(keys(parent_category.inductions)) 
+        return parent_category.inductions[X]
+    end
+
     @assert is_semisimple(parent(X)) "Requires semisimplicity"
     Z = direct_sum([s⊗X⊗dual(s) for s ∈ ind_simples])[1]
     a = associator
-    γ = Vector{Morphism}(undef, length(simples))
+    γ = Vector{Morphism}(undef, length(ind_simples))
+
     C = parent(X)
-    for i ∈ 1:length(simples)
-        W = simples[i]
+
+    for i ∈ 1:length(ind_simples)
+        W = ind_simples[i]
         γ[i] = zero_morphism(zero(parent(X)), direct_sum([W⊗((s⊗X)⊗dual(s)) for s ∈ ind_simples])[1])
 
         for S ∈ ind_simples
-            dom_i = ((S⊗X)⊗dual(S))⊗simples[i]
+            dom_i = ((S⊗X)⊗dual(S))⊗W
             γ_i_temp = zero_morphism(dom_i, zero(parent(X)))
             for  T ∈ ind_simples
                 #@show S,T,W
                 # Set up basis and dual basis
 
-                #basis, basis_dual = dual_basis(Hom(S, W⊗T), Hom(dual(S), dual(W⊗T)))
-                _basis, basis_dual = basis(Hom(S, W⊗T)), basis(Hom(dual(S)⊗W, dual(T)))
+                #_basis, basis_dual = dual_basis(Hom(S, W⊗T), Hom(dual(S), dual(W⊗T)))
+
+                #_basis, basis_dual = adjusted_dual_basis(Hom(S, W⊗T), Hom(dual(S)⊗W, dual(T)), S, W, T)
+
+                 _,_,ic,p = direct_sum_decomposition(W⊗T, simples)
+                _basis = [f for f ∈ ic if domain(f) == S]
+                dual_basis = [f for f ∈ p if codomain(f) == S]
+                #_,_,_,p = direct_sum_decomposition(dual(S)⊗W, dual.(simples))
+
+                basis_dual = transform_dual_basis(dual_basis, S,W,T)
 
                 if length(_basis) == 0 
                     γ_i_temp = vertical_direct_sum(γ_i_temp, zero_morphism(dom_i, W⊗((T⊗X)⊗dual(T)))) 
                     continue
                 end
 
-                corrections = base_ring(X).([(id(W)⊗ev(dual(T))) ∘ (id(W)⊗(spherical(T)⊗id(dual(T)))) ∘ a(W,T,dual(T)) ∘ (f⊗g) ∘ a(S,dual(S),W) ∘ (coev(S)⊗id(W)) for (f,g) ∈ zip(_basis,basis_dual)])
+
+                #_basis, basis_dual = adjusted_dual_basis(_basis, basis_dual, S, W, T)
+
+
+                #corrections = base_ring(X).([(id(W)⊗ev(dual(T))) ∘ (id(W)⊗(spherical(T)⊗id(dual(T)))) ∘ a(W,T,dual(T)) ∘ (f⊗g) ∘ a(S,dual(S),W) ∘ (coev(S)⊗id(W)) for (f,g) ∈ zip(_basis,basis_dual)])
 
+
                 #@show "component_iso"
-                component_iso = sum([a(W,T⊗X,dual(T)) ∘ (a(W,T,X)⊗id(dual(T))) ∘ ((f⊗id(X))⊗(inv(dim(W))*inv(k)*g)) ∘ a(S⊗X,dual(S),W) for (k,f,g) ∈ zip(corrections,_basis, basis_dual)])
+                #component_iso = sum([a(W,T⊗X,dual(T)) ∘ (a(W,T,X)⊗id(dual(T))) ∘ ((f⊗id(X))⊗(inv(dim(W))*inv(k)*g)) ∘ a(S⊗X,dual(S),W) for (k,f,g) ∈ zip(corrections,_basis, basis_dual)])
+
+                component_iso = sum([a(W,T⊗X,dual(T)) ∘ (a(W,T,X)⊗id(dual(T))) ∘ ((f⊗id(X))⊗(g)) ∘ a(S⊗X,dual(S),W) for (f,g) ∈ zip(_basis, basis_dual)])
+
                 #@show "sum"
                 γ_i_temp = vertical_direct_sum(γ_i_temp, (component_iso))
             end
-            γ[i] = horizontal_direct_sum(γ[i], dim(S)*γ_i_temp)
+            γ[i] = horizontal_direct_sum(γ[i], γ_i_temp)
         end
 
 
@@ -412,9 +441,19 @@ function partial_induction(X::Object, ind_simples::Vector, simples::Vector = sim
         γ[i] = inv(distr_after) ∘ γ[i] ∘ distr_before 
     end
 
-    return CenterObject(parent_category,Z,γ)
-end
+    if !is_split_semisimple(C)
+        # factor out such that the left and right tensor product coincide.
+        r = direct_sum([horizontal_direct_sum([image(f⊗id(X)⊗id(dual(b)) - id(b)⊗id(X)⊗dual(f))[2] for f in End(b)]) for b in simples])
+
+        @show Z,r = cokernel(r)
+        ir = right_inverse(r)
+
+        γ = [(id(b)⊗r) ∘ γᵢ ∘ (ir ⊗ id(b)) for (γᵢ,b) ∈ zip(γ, simples)]
+    end
 
+    IX = CenterObject(parent_category,Z,γ)
 
+    add_induction!(parent_category, X, IX)
 
-
+    return IX
+end