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BindGlobal("YBFamily", NewFamily("YBFamily"));
BindGlobal("YBType", NewType(YBFamily, IsYB));

### This function returns the set-theoretic solution given by the permutations <l_actions> and <r_actions>
### <l_action> and <r_action> are matrices!
InstallMethod(YB, "for a two lists of permutations", [ IsList, IsList ],
function(l, r)
  local obj;
  if not IS_YB(l, r) then
    Error("this is not a solution of the YBE");
  fi;

  obj := Objectify(YBType, rec());

  SetSize(obj, Size(l));
  SetLMatrix(obj, List(l, x->List(x, y->y)));
  SetRMatrix(obj, List(r, x->List(x, y->y)));

  return obj;

end);

InstallMethod(InverseYB, "for a solution", [ IsYB ],
function(obj)
  local x, y, n, t, l, u, v;
  t := Table(obj);
  n := Size(obj);
  l := NullMat(n,n);
  for x in [1..n] do
    for y in [1..n] do
      u := t[x][y][1];
      v := t[x][y][2];
      l[u][v] := [x,y];
    od;
  od;
  return Table2YB(l);
end);

InstallMethod(ViewObj,
  "for a set-theoretical solution",
  [ IsYB ],
  function(obj)
  Print("<A set-theoretical solution of size ", Size(obj), ">");
end);

InstallMethod(PrintObj,
  "for a set-theoretical solution",
  [ IsYB ],
  function(obj)
  if Size(obj) < 5 then
    Print("YB(", LMatrix(obj), ",", RMatrix(obj), ")");
  else
    Print("<A set-theoretical solution of size ", Size(obj), ">");
  fi;
end);

InstallMethod(SmallIYB, "for two integers", [ IsInt, IsInt ],
function(size, number)
  return CycleSet2YB(SmallCycleSet(size, number));
end);

InstallMethod(Permutations2YB, "for a list of permutations", [ IsList, IsList ],
function(l, r)
  return YB(List(l, x->ListPerm(x, Size(l))), List(r, y->ListPerm(y, Size(l))));
end);

### This function returns the set-theoretic solution
### corresponding to the matrix <table> such that the (i,j) entry is r(i,j)
InstallMethod(Table2YB, "for a square matrix", [ IsList ],
function(table)
  local ll, rr, x, y, p, n;

  n := Size(table);

  ll := List([1..n], x->[1..n]);
  rr := List([1..n], x->[1..n]);

  for x in [1..n] do
    for y in [1..n] do
      ll[x][y] := table[x][y][1];
      rr[y][x] := table[x][y][2];
    od;
  od;
  return YB(ll, rr);
end);

##### This function returns the table of the solution, which is
##### the matrix that in the (i,j)-entry has r(i,j)
InstallMethod(Table, "for a set theoretic solution", [ IsYB ],
function(obj)
  local m, x, y;
  m := NullMat(Size(obj), Size(obj));
  for x in [1..Size(obj)] do
    for y in [1..Size(obj)] do
      m[x][y] := TableYB(obj, x, y);
    od;
  od;
  return m;
end);

### This function returns true if <obj> is square-free
### A solution r is square-free iff r(x,x)=(x,x) for all x
InstallOtherMethod(IsSquareFree,
  "for a set-theoretical solution",
  [ IsYB ],
  function(obj)
  local x;
  for x in [1..Size(obj)] do
    if [LMatrix(obj)[x][x], RMatrix(obj)[x][x]] <> [x, x] then
      return false;
    fi;
  od;
  return true;
end);

InstallMethod(IYBGroup, "for a set-theoretical solution", [ IsYB ],
function(obj)
  return YBPermutationGroup(obj);
end);

InstallMethod(YBPermutationGroup, "for a set-theoretical solution", [ IsYB ],
function(obj)
  return Group(LPerms(obj));
end);

InstallMethod(Evaluate, "for a set-theoretical solution and a list of two elements", [ IsYB, IsList ],
function(obj, pair)
  local x,y;
  x := pair[1];
  y := pair[2];
  return [LMatrix(obj)[x][y], RMatrix(obj)[y][x]];
end);

### This function returns the vector value of <obj> at (<x>,<y>)
InstallMethod(TableYB, "for a set-theoretical solution", [ IsYB, IsInt, IsInt ],
function(obj, x, y)
  return [LMatrix(obj)[x][y], RMatrix(obj)[y][x]];
end);

### This function returns true if <obj> is involutive
### r is involutive <=> r^2=id
InstallMethod(IsInvolutive, "for a set-theoretical solution", [ IsYB ],
function(obj)
  local x,y,s;
  for x in [1..Size(obj)] do
    for y in [1..Size(obj)] do
      s :=  TableYB(obj, x, y);
      if TableYB(obj, s[1], s[2]) <> [x,y] then
        return false;
      fi;
    od;
  od;
  return true;
end);

#InstallMethod(YBTable, "for a set-theoretical solution", [ IsYB, IsInt, IsInt ],
#function(obj, i, j)
#  return [obj!.l_actions[x][y], obj!.r_actions[y][x]];
#end);

### This function returns true if the maps x->L_x are bijective
### r(x,y)=(L_x(y), R_y(x))
### CHECK
InstallMethod(IsLeftNonDegenerate,
  "for a set-theoretical solution",
  [ IsYB ],
  function(obj)
  local l;
  l := List(LMatrix(obj), PermList);
  if fail in l then
    return false;
  else
    SetLPerms(obj, l);
    return true;
  fi;

  #for x in [1..Size(obj)] do
  #  if PermList(LMatrix(obj)[x]) = fail then
  #    return false;
  #  fi;
  #od;
  #SetLPerms(obj, List(LMatrix(obj), PermList));
  #return true;
end);

### This function returns true if the maps x->R_x are bijective
### r(x,y)=(L_x(y), R_y(x))
InstallMethod(IsRightNonDegenerate,
  "for a set-theoretical solution",
  [ IsYB ],
  function(obj)
  local l;
  l := List(RMatrix(obj), PermList);
  if fail in l then
    return false;
  else
    SetRPerms(obj, l);
    return true;
  fi;
end);

### This function returns true if <r> is left and right non-degenerate
InstallMethod(IsNonDegenerate,
  "for a set-theoretical solution",
  [ IsYB ],
  function(obj)
  return IsLeftNonDegenerate(obj) and IsRightNonDegenerate(obj);
end);

InstallMethod(LPerms,
  "for a set-theoretical solution",
  [ IsYB ],
  function(obj)
  if IsLeftNonDegenerate(obj) then
    return LPerms(obj);# List(obj!.l_actions, x->PermList(x));
  fi;
  return fail;
end);

InstallMethod(RPerms,
  "for a set-theoretical solution",
  [ IsYB ],
  function(obj)
  if IsRightNonDegenerate(obj) then
    return RPerms(obj);
  fi;
  return fail;
end);

InstallMethod(YB2CycleSet,
  "for a set-theoretical solution",
  [ IsYB ],
  function(obj)

  if not IsInvolutive(obj) then
    Error("the solution of the YBE is not involutive");
  fi;

  if not IsRightNonDegenerate(obj) then
    Error("the soslution of the YBE is not (right) non-degenerate");
  fi;
  return CycleSet(List(RMatrix(obj), x->ListPerm(Inverse(PermList(x)),Size(obj))));

  return CycleSet(RMatrix(obj));
end);

InstallGlobalFunction(YB_xy,
  function(obj, x, y)
  return [LMatrix(obj)[x][y], RMatrix(obj)[y][x]];
  #return [obj!.l_actions[x][y], obj!.r_actions[y][x]];
end);

InstallMethod(Retract,
  "for a set-theoretical solution",
  [ IsYB ],
  function(obj)
  local e, c, s, pairs, x, y, z, ll, rr;

  if not IsInvolutive(obj) then
    Error("the solutions of the YBE is not involutive");
  fi;

  pairs := [];
  for x in [1..Size(obj)] do
    for y in [1..Size(obj)] do
      if LPerms(obj)[x] = LPerms(obj)[y] then
        Add(pairs, [x, y]);
      fi;
    od;
  od;

  e := EquivalenceRelationByPairs(Domain([1..Size(obj)]), pairs);
  c := EquivalenceClasses(e);
  s := Size(c);

  ll := List([1..s], x->[1..s]);
  rr := List([1..s], x->[1..s]);

  for x in [1..s] do
    for y in [1..s] do
      z := YB_xy(obj, Representative(c[x]), Representative(c[y]));
      ll[x][y] := Position(c, First(c, u->z[1] in u));
      rr[y][x] := Position(c, First(c, u->z[2] in u));
    od;
  od;
  return YB(ll, rr);
end);

InstallMethod(IsRetractable,
  "for a set-theoretical solution",
  [ IsYB ],
  function(obj)
  local r;
  r := Retract(obj);
  if Size(r) = Size(obj) then
    return false;
  else
    return true;
  fi;
end);

InstallMethod(IsMultipermutation,
  "for a set-theoretical solution",
  [ IsYB ],
  function(obj)
  if not MultipermutationLevel(obj) = fail then
    return true;
  else
    return false;
  fi;
end);

InstallMethod(MultipermutationLevel,
  "for a set-theoretical solution",
   [ IsYB ],
  function(obj)
  local r,s,l;

  l := 0;
  r := ShallowCopy(obj);

  repeat
    s := ShallowCopy(r);
    r := Retract(s);
    if Size(r) <> Size(s) then
      l := l+1;
    else
      return fail;
    fi;
  until Size(r) = 1;
  return l;
end);

### This function returns the number of involutive set-theoretic solutions of size <n>
InstallGlobalFunction(NrSmallIYB,
function(n)
  return NrSmallCycleSets(n);
end);

### This function returns the permutations defining the set-theoretic solution
### These permutations are in the following form: [ left_permutations, right_permutations ]
InstallOtherMethod(Permutations,
  "for set-theoretic solutions",
  [ IsYB ],
  function(obj)
  return [LPerms(obj), RPerms(obj)];
end);

### This function returns the <i>-th involutive set-theoretic solution of size <n>
### These solutions were computed by Etingof, Schedler and Soloviev
#InstallGlobalFunction(SmallIYB,
#function(n, i)
#  return CycleSet2YB(SmallCycleSet(n,i));
#  local r, data;
#  data := [
#    YB_size1,
#    YB_size2,
#    YB_size3,
#    YB_size4,
#    YB_size5,
#    YB_size6,
#    YB_size7,
#    YB_size8
#  ];
#  if n in [1..8] then
#    r := data[n][i];
#    return YB(r[1], r[2]);
#  else
#    return fail;
#  fi;
#end);

InstallGlobalFunction(YB_IsBraidedSet,
function(l_actions, r_actions)
  local x, y, z, v;

  if Size(r_actions) <> Size(l_actions) then
    return false;
  fi;

  for x in [1..Size(l_actions)] do
    for y in [1..Size(l_actions)] do
      for z in [1..Size(l_actions)] do
        v := [x,y,z];
        if YB_ij(l_actions, r_actions, YB_ij(l_actions, r_actions, YB_ij(l_actions, r_actions, v, 2, 3), 1, 2), 2, 3) \
          <> YB_ij(l_actions, r_actions, YB_ij(l_actions, r_actions, YB_ij(l_actions, r_actions, v, 1, 2), 2, 3), 1, 2) then
          return false;
        fi;
      od;
    od;
  od;
  return true;
end);

### This function returns true if <subset> is r-invariant
### A subset Y of X is r-invariant if r(YxY) is included in YxY
InstallMethod(IsInvariant, "for a solution and a list", [IsYB, IsList],
function(obj, subset)
  local x, y, z;
  for x in subset do
    for y in subset do
      z := YB_xy(obj, x, y);
      if not z[1] in subset or not z[2] in subset then
        return false;
      fi;
    od;
  od;
  return true;
end);

### This function returns the restricted solution with respect to <subset>
### If <subset> is not invariant, the function returns fail
InstallMethod(RestrictedYB, "for a solution and a list", [ IsYB, IsList ],
function(obj, subset)
  local x, y, z, ll, rr;

  if not IsInvariant(obj, subset) then
    return fail;
  fi;

  ll := List([1..Size(subset)], x->[1..Size(subset)]);
  rr := List([1..Size(subset)], x->[1..Size(subset)]);

  for x in [1..Size(subset)] do
    for y in [1..Size(subset)] do
      ll[x][y] := Position(subset, subset[y]^LPerms(obj)[subset[x]]);
      rr[y][x] := Position(subset, subset[x]^RPerms(obj)[subset[y]]);
    od;
  od;
  return YB(ll, rr);
end);

### This function returns the structure group of <r>
### Generators: x_1,x_2,...,x_n
### Relations: x_ix_j=x_kx_l whenever r(i,j)=(k,l)
InstallMethod(StructureGroup, "for a solution", [ IsYB ],
function(obj)
  local n, f, x, y, rels, i, j;

  n := Size(obj);
  f := FreeGroup(n);
  x := GeneratorsOfGroup(f);

  rels := [];

  for i in [1..n] do
    for j in [1..n] do
      y := YB_xy(obj, i, j);
      Add(rels, x[i]*x[j]*Inverse(x[y[1]]*x[y[2]]));
    od;
  od;
  return f/rels;
end);

### This function returns the trivial solution over [1..size]
### r is trivial <=> r(x,y)=(y,x)
InstallMethod(TrivialYB, "for an integer", [ IsInt ],
function(n)
  return YB(List([1..n], x->[1..n]), List([1..n], x->[1..n]));
end);

InstallMethod(LyubashenkoYB, "for an integer and two permutations", [ IsInt, IsPerm, IsPerm ],
function(size, f, g)
  if f*g=g*f then
    return YB(List([1..size], x->ListPerm(f, size)), List([1..size], x->ListPerm(g,size)));
  else
    return fail;
  fi;
end);

### This function returns the cartesian product of the solutions <r> and <s>
### This means: r((a,b),(c,d))=(r(a,c),r(b,d))
InstallMethod(CartesianProduct, "for two solutions", [ IsYB, IsYB ],
function(r, s)
  local c, l, u, v, w;
  l := [];
  c := Cartesian([1..Size(r)], [1..Size(s)]);
  for u in c do
    for v in c do
      w := [YB_xy(r, u[1], v[1]),  YB_xy(s, u[2], v[2])];
      Add(l, [[Position(c, u), Position(c, v)], [Position(c, [w[1][1], w[2][1]]), Position(c, [w[1][2], w[2][2]])]]);
    od;
  od;
  return Table2YB(l);
end);

### This function returns the matrix of the rack corresponding to the derived solution of a solution
InstallMethod(DerivedRack, "for a solution", [ IsYB ],
function(obj)
  local x, y, z, m;

  if not IsNonDegenerate(obj) then
    return fail;
  fi;

  m := NullMat(Size(obj), Size(obj));
  for x in [1..Size(obj)] do
    for y in [1..Size(obj)] do
      # I have two operations. Let S(x,y)=(g_x(y),f_y(x)) and
      # xoy=g_y(x), y*x=Inverse(f)_y(x).
      # Then the derived rack structure is given by x>y=f_x((y*x)oy)
      z := y^LPerms(obj)[x^Inverse(RPerms(obj)[y])];
      m[x][y] := z^RPerms(obj)[x];
    od;
  od;
  return Rack(m);
end);

InstallMethod(Wada, "for a group", [ IsGroup ],
function(group)
  local x, y, e, l, r;

  e := Elements(group);
  l := NullMat(Size(group), Size(group));
  r := NullMat(Size(group), Size(group));
  for x in group do
    for y in group do
      l[Position(e, x)][Position(e, y)] := Position(e, x*Inverse(y)*Inverse(x));
      r[Position(e, y)][Position(e, x)] := Position(e, x*y^2);
    od;
  od;
  return YB(l, r);
end);

### CHECK and FIXME
InstallMethod(IsBiquandle, "for a solution", [ IsYB ],
function(obj)
  local x, y;
  for x in [1..Size(obj)] do
    if not ForAny([1..Size(obj)], y->YB_xy(obj, x, y)=[x,y]) then
      return false;
    fi;
  od;
  return true;
end);

InstallMethod(YB2Permutation, "for a solution", [ IsYB ],
function(obj)
  local perm, x, y, u, v;

  perm := [1..Size(obj)^2];

  for x in [1..Size(obj)] do
    for y in [1..Size(obj)] do
      u := YB_xy(obj, x, y)[1];
      v := YB_xy(obj, x, y)[2];
      perm[x+Size(obj)*y] := u+Size(obj)*v;
    od;
  od;
  return PermList(perm);
end);

## j>i = s_y t_(s_x^(-1)(y))(x)
#lrack := function(lperms, rperms)
#  local i,j,m,n;
#  n := Size(lperms);
#  m := NullMat(n,n);
#  for i in [1..n] do
#    for j in [1..n] do
#      m[j][i] := (i^lperms[j^Inverse(rperms[i])])^rperms[j];
#    od;
#  od;
#  return Rack(m);
#end;

InstallMethod(DerivedRightRack, "for a solution", [ IsYB ],
function(obj)
  local i,j,m,n,lperms,rperms;
  lperms := List(obj!.l_actions, PermList);
  rperms := List(obj!.r_actions, PermList);
  n := Size(lperms);
  m := NullMat(n,n);
  for i in [1..n] do
    for j in [1..n] do
      m[j][i] := (i^rperms[j^Inverse(lperms[i])])^lperms[j];
    od;
  od;
  return Rack(m);
end);

# j>i = s_y t_(s_x^(-1)(y))(x)
# it follows the notation of my paper with Lebed
InstallMethod(DerivedLeftRack, "for a solution", [ IsYB ],
function(obj)
  return DerivedRack(obj);
end);

# FIXME: IsObject?
InstallMethod(DehornoyRepresentationOfStructureGroup, "for an involutive solution and a symbol", [ IsYB, IsObject ],
function(obj,q)
  local i, m, gens;

  gens := [];

if not IsInvolutive(obj) then
  return fail;
  fi;

  for i in [1..Size(obj)] do
    m := One(Field(q))*IdentityMat(Size(obj), Size(obj));
    m[i][i] := q;
    Add(gens, m*PermutationMat(Inverse(Permutations(obj)[1][i]), Size(obj)));
  od;

  return Group(gens);
end);

InstallMethod(DehornoyClass, "for an involutive solution", [ IsYB ],
function(obj)
  local x, m, tmp, sigmas, T;

if not IsInvolutive(obj) then
  return fail;
fi;

  tmp := [];
  sigmas := Permutations(obj)[2];
T := PermList(List([1..Size(obj)], x->x^Inverse(sigmas[x])));

  for x in [1..Size(obj)] do
    Add(tmp, First(PositiveIntegers, m->Product(Reversed(List([0..m-1], k->sigmas[x^(T^k)]))) = ()));
  od;
  return Lcm(tmp);
end);

InstallMethod(IdYB, "for an involutive solution", [ IsYB ],
function(obj)
  local cs;
  cs := YB2CycleSet(obj);
  if not Size(obj) in [1..8] then
    Error("solutions of size ", Size(obj), " are not in the database");
  else
    return [Size(obj), First([1..NrSmallCycleSets(Size(obj))], k->IsomorphismCycleSets(cs, SmallCycleSet(Size(obj),k)) <> fail)];
  fi;
end);

InstallMethod(IYBBrace, "for an involutive solution", [ IsYB ],
function(obj)
  local gens, fam, brace, sum, group, add, x, y, mul, ESS_1cocycle, AllMatrices, ESS_inverse1cocycle;

group := AffineCrystGroupOnLeft(GeneratorsOfGroup(LinearRepresentationOfStructureGroup(obj)));

  ESS_1cocycle := function(m)
   local n, A;
   n := Size(m);
   A := TransposedMat(m);
   A := A{[n]}{[1..(n-1)]};
   return A[1];
  end;

  AllMatrices := function(hol,v)
   local n, P, M, i, z, V, A, B;
   n := Size(v);
   P := AsList(hol);

   A := [];
   B := [];
   z := NullMat(n,n);

   for i in [1..n] do
      z[i][1]:=v[i];
   od;

   B[1] := BlockMatrix([ [1,1, P[1] ], [1,2, z], [2,2,One(P[1])]],2,2);
   A[1] := MatrixByBlockMatrix(B[1]);
   A[1] := A[1]{[1..n+1]}{[1..n+1]};

    for i in [2..Size(P)] do
    B[i]:=BlockMatrix([[1,1,P[i]], [1,2,z], [2,2,One(P[i])]],2,2);
    A[i]:=MatrixByBlockMatrix(B[i]);
      A[i]:=A[i]{[1..n+1]}{[1..n+1]};
    od;
   return A;
  end;

  ESS_inverse1cocycle := function(v)
   local hol;
   hol := PointGroup(group);
    return First(AllMatrices(hol, v), x->x in group);
  end;

  sum := function(x, y)
  local f,g,h,a,b,c;

  f := PointHomomorphism(group);

  g := PreImagesRepresentative(f,x);
  h := PreImagesRepresentative(f,y);

  a:=ESS_1cocycle(g);
  b:=ESS_1cocycle(h);

  return Image(f, ESS_inverse1cocycle(a+b));

end;

mul := AsList(PointGroup(group));
add := NullMat(Size(mul),Size(mul));

for x in mul do
  for y in mul do
   add[Position(mul, x)][Position(mul,y)] := Position(mul, sum(x,y));
  od;
od;

add := List(add, PermList);

  fam := NewFamily("SkewbraceElmFamily", IsSkewbraceElm, IsMultiplicativeElementWithInverse and IsAdditiveElementWithInverse);
  fam!.DefaultType := NewType(fam, IsSkewbraceElmRep);
  brace := Objectify(NewType(CollectionsFamily(fam), IsSkewbrace and IsAttributeStoringRep), rec());
  fam!.Skewbrace := brace;

  SetSize(brace, Size(add));
  SetSkewbraceAList(brace, add);
  SetSkewbraceMList(brace, mul);

  #gens := SmallGeneratingSet(Group(add));
  gens := GeneratorsOfGroup(Group(add));
  if gens = [] then
    SetUnderlyingAdditiveGroup(brace, Group(()));
  else
    SetUnderlyingAdditiveGroup(brace, Group(gens));
  fi;

  gens := GeneratorsOfGroup(Group(mul));
  if gens = [] then
    SetUnderlyingMultiplicativeGroup(brace, Group(()));
  else
    SetUnderlyingMultiplicativeGroup(brace, Group(gens));
  fi;

  return brace;

end);

InstallMethod(LinearRepresentationOfStructureGroup, "for an involutive solution", [ IsYB ],
function(obj)
  local i,j,m,l,n,x,perms;

  if not IsInvolutive(obj) then
    return fail;
  fi;

  n := Size(obj);
  l := [];
  perms := GeneratorsOfGroup(DehornoyRepresentationOfStructureGroup(obj,1));

  for x in [1..n] do

    m := NullMat(n+1, n+1);

    for i in [1..n] do
      for j in [1..n] do
        m[i][j] := perms[x][i][j];
      od;
    od;

    m[x][n+1] := 1;
    m[n+1][n+1] := 1;

    Add(l, m);

  od;

  return Group(l);

end);

InstallMethod(IsIndecomposable,
  "for cycle sets",
  [ IsYB ],
  function(obj)
    if not IsInvolutive(obj) then
      Error("the solution of the YBE is not involutive");
    else
      return IsTransitive(IYBGroup(obj), [1..Size(obj)]);
    fi;
end);

Impressum ybe.gi Interaktion und Portierbarkeitunbekannt

[ Seitenstruktur0.42Drucken etwas mehr zur Ethik ]

Impressum ybe.gi Interaktion und
Portierbarkeitunbekannt