Add anyon wiki

src/AnyonWiki/AnyonWiki.jl

@@ -0,0 +1,139 @@
+#=----------------------------------------------------------
+    Load fusion categories from the Anyon Wiki by 
+    Gert Vercleyen. 
+----------------------------------------------------------=#
+
+associator_path = joinpath(@__DIR__, "MultFreeCenters/6jSymbols/")
+pivotal_path = joinpath(@__DIR__, "MultFreeCenters/PivotalStructures/")
+
+@doc raw""" 
+
+    anyonewiki(n::Int64)
+
+Load the n-th fusion category from the list of multiplicity free fusion categories of rank ≤ 7.
+"""
+function anyonwiki(n::Int)
+
+    data = []
+    CC = AcbField(512)
+    Q = QQBarField()
+    K = load_anyon_number_field(n::Int)
+    deg = degree(K)
+    e = complex_embeddings(K)[1]
+
+    # Open File n and extract data
+    open(joinpath(associator_path, "cat_$n")) do f 
+
+        while ! eof(f)
+
+            s = readline(f) 
+            s = filter(!=('\"'), s)
+            s = split(s, " ")
+
+            indices = Int[eval(Meta.parse(a)) for a ∈ s[1:end-3]]
+            x_real = CC(s[end-2] * " +/- 1e-512")
+            x_imag = CC(s[end][1:end-2] * "+/- 1e-512") * CC(im) 
+
+            x = guess(Q, x_real + x_imag, deg)
+
+            append!(data, [Any[indices; preimage(e,x)]])
+        end
+    end
+    N = maximum([a[1] for a ∈ data])
+
+    # Construct multiplication table from associators
+    mult = zeros(Int,N,N,N)
+
+    for (_,a,b,c) ∈ filter(e -> e[1] == 1, data) 
+
+        mult[a,b,c] = 1 
+    end
+
+    C = six_j_category(K, mult)
+    set_name!(C, "Fusion category number $n from AnyonWiki List")
+
+    # Build matrices for 6j-Symbols
+    for a ∈ 1:N, b ∈ 1:N, c ∈ 1:N, d ∈ 1:N
+        abc_d = filter(e -> e[[1,2,3,4]] == [a,b,c,d], data)
+        l = Int(sqrt(length(abc_d)))
+        M = matrix(K,l,l, [v[7] for v ∈ abc_d])
+        set_associator!(C,a,b,c,d,M)  
+    end
+    set_one!(C, 1)
+
+    set_pivotal!(C, load_anyon_pivotal(n, K, e))
+
+    C
+end
+
+function load_anyon_number_field(n::Int)
+    line = 1
+    open(joinpath(associator_path, "Polynomials")) do f 
+
+        while ! eof(f) 
+
+            s = readline(f) 
+
+            if line == n 
+                global _,x = QQ[:x]
+                min_pol = eval(Meta.parse(s))
+
+                global K1 = min_pol == 1 ? QQ : number_field(min_pol)[1]
+                break
+            end
+
+            line += 1
+        end
+    end
+
+    line = 1
+    open(joinpath(pivotal_path, "Polynomials")) do f 
+
+        while ! eof(f) 
+
+            s = readline(f) 
+
+            if line == n 
+                global _,x = QQ[:x]
+                min_pol = eval(Meta.parse(s))
+
+                global K2 =  min_pol == 1 ? QQ : number_field(min_pol)[1]
+                break
+            end
+
+            line += 1
+        end
+    end
+
+    K1 == K2 && return K1
+    K1 == QQ && return K2 
+    K2 == QQ && return K1 
+
+    simplify(splitting_field([defining_polynomial(K1), defining_polynomial(K2)]))[1]
+end
+
+function load_anyon_pivotal(n::Int, K = load_anyon_number_field(n), e = complex_embeddings(K)[1])
+
+    CC = AcbField(512)
+    Q = QQBarField()
+    deg = degree(K) 
+
+    piv = elem_type(K)[]
+    open(joinpath(pivotal_path, "pivots_cat_$n")) do f 
+
+        while ! eof(f) 
+
+            s = readline(f) 
+            s = filter(!=('\"'), s)
+            s = split(s, " ")
+
+            x_real = CC(s[2] * " +/- 1e-512")
+            x_imag = CC(s[4][1:end-2] * "+/- 1e-512") * CC(im) 
+
+            x = guess(Q, x_real + x_imag, deg)
+
+            append!(piv, [preimage(e, x)])
+        end
+    end
+    return piv 
+end

src/CategoryFramework/AbstractMethods.jl

@@ -503,6 +503,8 @@ end
 # end
 
 
+rank(C::Category) = length(simples(C))
+
 is_isomorphic_simples(X::Object, Y::Object) = is_isomorphic(X,Y)
 
 

src/Examples/TambaraYamagami/TambaraYamagami.jl

@@ -115,7 +115,7 @@ function tambara_yamagami(K::Ring, A::Group, τ::RingElem, χ::BilinearForm)
     end
     set_associator!(TY, n+1, n+1, n+1, [[zero_mat for _ ∈ 1:n]; inv(a)*matrix(K,[inv(χ(els[i],els[j])) for i ∈ 1:n, j ∈ 1:n])])
     set_one!(TY, [1; [0 for _ ∈ 1:n]])
-    set_spherical!(TY, [K(1) for _ ∈ 1:n+1])
+    set_pivotal!(TY, [K(1) for _ ∈ 1:n+1])
 
     set_attribute!(TY, :classification, ["Tambara-Yamagami"])
 
@@ -262,7 +262,7 @@ function Ising(F::Ring, sqrt_2::RingElem, q::Int)
 
     set_one!(C,[1,0,0])
 
-    set_spherical!(C, [F(1) for s ∈ simples(C)])
+    set_pivotal!(C, [F(1) for s ∈ simples(C)])
 
     set_attribute!(C, :classification, ["Ising", "Tambara-Yamagami"])
 

src/Examples/VercleyenSingerland/FR_9143/fr_9143.jl

@@ -71,6 +71,6 @@ function cat_fr_9143()
         end
     end    
     set_one!(C,[1,0,0,0,0,0,0,0,0])
-    TensorCategories.set_spherical!(C, [K(1) for s ∈ simples(C)])
+    TensorCategories.set_pivotal!(C, [K(1) for s ∈ simples(C)])
     return C
 end

src/Examples/Verlinde/I2-fusion.jl

@@ -190,7 +190,7 @@ function I2subcategory(m, F::Ring = cyclotomic_field(2*m)[1])
 	A=[0 for s in simples(C)]::Vector{Int64}
 	A[1]=1
 	set_one!(C,A)
-	set_spherical!(C, [F(1) for s ∈ simples(C)])
+	set_pivotal!(C, [F(1) for s ∈ simples(C)])
 	set_name!(C, "Fusion subcategory of I₂($m)")
 	return C
 end

src/SixJCategoryDatabase/main.jl

@@ -6,8 +6,9 @@ function load_fusion_category(name::String)
     load(joinpath(@__DIR__, name * ".mrdi"))
 end
 
-function add_to_local_database(C::SixJCategory, name::String)
-    files = readdir(joinpath(@__DIR__))
+function add_to_local_database(C::SixJCategory, name::String, path::String = @__DIR__)
+
+    files = readdir(joinpath(path))
 
     if name * ".mrdi" ∈ files 
         println("File already existent. Do you want to overwrite? [y/n]")
@@ -20,5 +21,5 @@ function add_to_local_database(C::SixJCategory, name::String)
             return 
         end
     end
-    save(joinpath(@__DIR__, name * ".mrdi"), C)
+    save(joinpath(path, name * ".mrdi"), C)
 end

src/TensorCategories.jl

@@ -12,7 +12,7 @@ import Oscar: +, @alias, @attributes, AbstractSet, AcbField, StructureConstantAl
     Hecke.AbstractAssociativeAlgebraElem, Ideal, MPolyRingElem, MPolyIdeal, Map, MatElem, MatrixElem,
     MatrixGroup, matrix_space, ModuleIsomorphism, number_field, PcGroup, PolyRingElem,
     polynomial_ring, QQ, QQBarField, QQField, QQFieldElem, QQMPolyRingElem, Ring, RingElem, ZZ, QQBarFieldElem,
-    ZZRingElem, abelian_closure, abelian_group, absolute_simple_field, action, base_field,
+    ZZRingElem, abelian_closure, abelian_group, absolute_simple_field, action, base_field, defining_polynomial,
     base_ring, basis, central_primitive_idempotents, change_base_ring, 
      characteristic,
     charpoly, codomain, coeff, coefficients, cokernel, complex_embeddings, compose, centralizer, embedding,
@@ -86,6 +86,7 @@ export algebras
 export algebra_structures
 export AlgebraObject
 export AlgebraMorphism
+export anyonwiki
 export ArrowCategory
 export ArrowObject
 export ArrowMorphism
@@ -325,6 +326,7 @@ export pairing
 export parent 
 export pentagon_axiom 
 export pentagon_equations
+export pivotal
 export preimage
 export print_multiplication_table 
 export print_multiplication_table 
@@ -378,7 +380,8 @@ export set_braiding!
 export set_canonical_spherical! 
 export set_name!
 export set_one! 
-export set_spherical! 
+export set_pivotal! 
+export set_spherical!
 export set_tensor_product! 
 export set_trivial_spherical!
 export SetHomSet 
@@ -515,7 +518,7 @@ include("Examples/Haagerup/ExtendedHaagerup.jl")
 
 include("Serialization/SixJSerialization.jl")
 include("SixJCategoryDatabase/main.jl")
-
+include("AnyonWiki/AnyonWiki.jl")
 
 
 

src/TensorCategoryFramework/AbstractTensorMethods.jl

@@ -154,6 +154,8 @@ spherical of a morphism ``X → X``.
 """
 tr(f::Morphism) = left_trace(f)
 
+pivotal(X::Object) = spherical(X::Object)
+
 """ 
 
     left_trace(f::Morphism)
@@ -169,7 +171,7 @@ function left_trace(f::Morphism)
     if V == zero(C) || W == zero(C) return zero_morphism(one(C),one(C)) end
 
     if V == W
-        return ev(left_dual(V)) ∘ ((spherical(V)∘f) ⊗ id(left_dual(V))) ∘ coev(V)
+        return ev(left_dual(V)) ∘ ((pivotal(V)∘f) ⊗ id(left_dual(V))) ∘ coev(V)
     end
     return ev(left_dual(V)) ∘ (f ⊗ id(left_dual(V))) ∘ coev(V)
 end

src/TensorCategoryFramework/Center/Center.jl

@@ -115,7 +115,7 @@ Return the image under the forgetful functor.
 """
 morphism(f::CenterMorphism) = f.m
 
-is_weakly_fusion(C::CenterCategory) = true
+is_weakly_fusion(C::CenterCategory) = dim(category(C)) != 0
 is_fusion(C::CenterCategory) = all([int_dim(End(s)) == 1 for s ∈ simples(C)])
 is_abelian(C::CenterCategory) = true
 is_linear(C::CenterCategory) = true
@@ -1022,7 +1022,13 @@ function add_induction!(C::CenterCategory, X::Object, IX::CenterObject)
 end
 
 function simples_by_induction!(C::CenterCategory, log = true)
-    S = CenterObject[]
+    S = CenterObject[one(C)]
+
+    if rank(category(C)) == 1
+        C.simples = S
+        return S 
+    end
+
     d = dim(C.category)^2
     C.induction_gens = induction_generators(C)
     simpls = simples(C.category)
@@ -1147,6 +1153,25 @@ function split(X::CenterObject, E = End(X))
     collect(values(eigenvalues(f ⊗ L)))
 end
 
+function split(C::CenterCategory)
+    ends = End.(simples(C))
+
+    non_split = filter(e -> int_dim(e) > 1, ends)
+
+    length(non_split) == 0 && return C 
+
+    minpolys = [minpoly.(basis(e)) for e ∈ non_split]
+
+    K = base_ring(C) 
+    _,x = K[:x]
+    minpolys = [B[findfirst(f -> degree(f) == length(B), B)](x) for B ∈ minpolys]
+
+    L = simplify(absolute_simple_field(splitting_field(minpolys))[1])[1]
+
+    incl = K == QQ ? L : is_subfield(K,L)[2]
+    extension_of_scalars(C, L, incl)
+end
+
 
 #=----------------------------------------------------------
     Hom Spaces 2.0