version="$Id: involut.lib,v 1.3 2005-04-27 19:32:14 levandov Exp $";
category="Noncommutative";
info="
LIBRARY:  involution.lib  Procedures for Computations and Operations with Involutions 
AUTHORS:  Oleksandr Iena,       yena@mathematik.uni-kl.de,
@*        Markus Becker,        mbecker@mathematik.uni-kl.de,
@*        Viktor Levandovskyy,  levandov@mathematik.uni-kl.de

SUPPORT: Forschungsschwerpunkt 'Mathematik und Praxis' (Project of Dr. E. Zerz
and V. Levandovskyy), Uni Kaiserslautern

NOTE: This library provides algebraic tools for computations and operations
with algebraic involutions and linear automorphisms of noncommutative algebras

PROCEDURES:
find_invo();          describes a variety of linear involutions on a basering;
find_invo_diag();     describes a variety of homothetic (diagonal) involutions on a basering;
find_auto();          describes a variety of linear automorphisms of a basering;
ncdetection(ring r);  computes an ideal, presenting an involution map on some particular noncommutative algebras;
involution(m, map theta);  applies the involution to an object.
";

LIB "ncalg.lib";
LIB "poly.lib";
LIB "primdec.lib";
///////////////////////////////////////////////////////////////////////////////
proc ncdetection(def r)
"USAGE:  ncdetection(r), r a ring
PURPOSE: compute optimized involutions for some particular noncommutative algebras
RETURN:  ideal (presenting an involution map)
NOTE:    the procedure is aimed at noncommutative algebras with differential, shift or advance operators arising in the Control Theory
EXAMPLE: example ncdetection; shows an example
"{
// in this procedure an involution map is generated from the NCRelations 
// that will be used in the function involution
// in dieser proc. wird eine matrix erzeugt, die in der i-ten zeile die indices
// der differential-, shift- oder advance-operatoren enthaelt mit denen die i-te
// variable nicht kommutiert. 
  int i,j,k,LExp;
  int NVars  = nvars(r);
  matrix rel = NCRelations(r)[2];
  intmat M[NVars][3];
  int NRows = nrows(rel);
  intvec v,w; 
  poly d,d_lead;
  ideal I;
  map theta;
  for( j=NRows; j>=2; j-- )
  {
   if( rel[j] == w )       //the whole column is zero
    {
      j--;
      continue;
    }
    for( i=1; i<j; i++ )         
    {
      if( rel[i,j]==1 )        //relation of type var(j)*var(i) = var(i)*var(j) +1
      {
         M[i,1]=j;
      }
      if( rel[i,j] == -1 )    //relation of type var(i)*var(j) = var(j)*var(i) -1
      {
        M[j,1]=i;
      }
      d = rel[i,j];
      d_lead = lead(d);
      v = leadexp(d_lead); //in the next lines we check wether we have a  relation of differential or shift type
      LExp=0;
      for(k=1; k<=NVars; k++)
      {
        LExp = LExp + v[k];
      }
      //      if( (d-d_lead != 0) || (LExp > 1) )
if ( ( (d-d_lead) != 0) || (LExp > 1) || ( (LExp==0) && ((d_lead>1) || (d_lead<-1)) ) )
      {
        return(theta);
      }

      if( v[j] == 1)                   //relation of type var(j)*var(i) = var(i)*var(j) -lambda*var(j)
      {
        if (leadcoef(d) < 0)
        {
          M[i,2] = j;
        }
        else
        {
          M[i,3] = j;
        }
      }
      if( v[i]==1 )                    //relation of type var(j)*var(i) = var(i)*var(j) -lambda*var(i)
      {
        if (leadcoef(d) > 0)
        {
          M[j,2] = i;
        }
        else
        {
          M[j,3] = i;
        }
      }
    }
  }
  // from here on, the map is computed
  for(i=1;i<=NVars;i++)
  {
    I=I+var(i);
  }

  for(i=1;i<=NVars;i++)
  {
    if( M[i,1..3]==(0,0,0) )
    {
      i++;
      continue;
    }
    if( M[i,1]!=0 )
    {
      if( (M[i,2]!=0) && (M[i,3]!=0) )
      {
        I[M[i,1]] = -var(M[i,1]);
        I[M[i,2]] = var(M[i,3]);
        I[M[i,3]] = var(M[i,2]);
      }
      if( (M[i,2]==0) && (M[i,3]==0) )
      {
        I[M[i,1]] = -var(M[i,1]);
      }                  
      if( ( (M[i,2]!=0) && (M[i,3]==0) )|| ( (M[i,2]!=0) && (M[i,3]==0) ) 
)
      {
        I[i] = -var(i);
      }
    }
    else
    {
      if( (M[i,2]!=0) && (M[i,3]!=0) )
      {
        I[i] = -var(i);
        I[M[i,2]] = var(M[i,3]);
        I[M[i,3]] = var(M[i,2]);
      }
      else
      {
        I[i] = -var(i);
      } 
    }
  }
  return(I);
}
example
{ 
  "EXAMPLE:"; echo = 2;
  ring r=0,(x,y,z,D(1..3)),dp;
  matrix D[6][6];
  D[1,4]=1; D[2,5]=1;  D[3,6]=1;
  ncalgebra(1,D);
  ncdetection(r);
  kill r;
  //----------------------------------------
  ring r=0,(x,S),dp;
  ncalgebra(1,-S);
  ncdetection(r);
  kill r;
  //----------------------------------------
  ring r=0,(x,D(1),S),dp;
  matrix D[3][3];
  D[1,2]=1;  D[1,3]=-S;
  ncalgebra(1,D);
  ncdetection(r);
}

static proc In_Poly(poly mm, list l, int NVars)
// applies the involution to the poly mm
// entries of a list l are images of variables under invo
// more general than invo_poly; used in many rings setting
{
  int i,j;
  intvec v;
  poly pp, zz;
  poly nn = 0;
  i = 1;
  while(mm[i]!=0)
  {
    v  = leadexp(mm[i]);
    zz = 1;
    for( j=NVars; j>=1; j--)
    {
      if (v[j]!=0)
      { 
        pp = l[j];
        zz = zz*(pp^v[j]);
      }
    }    
    nn = nn + (leadcoef(mm[i])*zz);
    i++;  
  }
  return(nn);
}

static proc Hom_Poly(poly mm, list l, int NVars)
// applies the endomorphism to the poly mm
// entries of a list l are images of variables under endo
// should not be replaced by map-based stuff! used in
// many rings setting
{
  int i,j;
  intvec v;
  poly pp, zz;
  poly nn = 0;
  i = 1;
  while(mm[i]!=0)
  {
    v  = leadexp(mm[i]);
    zz = 1;
    for( j=NVars; j>=1; j--)
    {
      if (v[j]!=0)
      { 
        pp = l[j];
        zz = (pp^v[j])*zz;
      }
    }    
    nn = nn + (leadcoef(mm[i])*zz);
    i++;  
  }
  return(nn);
}

static proc invo_poly(poly m, map theta)
// applies the involution map theta to m, where m=polynomial
{
  // compatibility:
  ideal l = ideal(theta);
  int i;
  list L;
  for (i=1; i<=size(l); i++)
  {
    L[i] = l[i];
  }
  int nv = nvars(basering);
  return (In_Poly(m,L,nv));
//   if (m==0) { return(m); }
//   int i,j;
//   intvec v;
//   poly p,z;
//   poly n = 0;
//   i = 1;
//   while(m[i]!=0)
//   {
//     v = leadexp(m[i]);
//     z =1;
//     for(j=nvars(basering); j>=1; j--)
//     {
//       if (v[j]!=0)
//       { 
//         p = var(j); 
//         p = theta(p);
//         z = z*(p^v[j]);
//       }
//     }    
//     n = n + (leadcoef(m[i])*z);
//     i++;  
//   }
//   return(n);
}
///////////////////////////////////////////////////////////////////////////////////
proc involution(m, map theta)
"USAGE:  involution(m, theta); m is a poly/vector/ideal/matrix/module, theta is a map
PURPOSE: applies the involution, presented by theta to the input m
RETURN:  object of the same type as m
EXAMPLE: example involution; shows an example
"{
  // applies the involution map theta to m, 
  // where m= vector, polynomial, module, matrix, ideal
  int i,j;
  intvec v;
  poly p,z;
  if (typeof(m)=="poly")
  {
    return (invo_poly(m,theta));  
  }
  if ( typeof(m)=="ideal" )
  {
    ideal n;
    for (i=1; i<=size(m); i++)
    {
      n[i] = invo_poly(m[i], theta);
    } 
    return(n);
  }
  if (typeof(m)=="vector")
  {
    for(i=1; i<=size(m); i++)
    {
      m[i] = invo_poly(m[i], theta);
    }
    return (m);  
  }
  if ( (typeof(m)=="matrix") || (typeof(m)=="module"))
  {  
    matrix n = matrix(m);
    int @R=nrows(n);
    int @C=ncols(n);
    for(i=1; i<=@R; i++)
    {
      for(j=1; j<=@C; j++)
      {
	if (m[i,j]!=0)
	{
	  n[i,j] = invo_poly( m[i,j], theta);
	}
      }
    }
    if (typeof(m)=="module")
    {
      return (module(n));
    }
    else // matrix
    {
      return(n);
    }
  }
  // if m is not of the supported type:
  "Error: unsupported argument type!";
  return();
}
example
{ 
  "EXAMPLE:";echo = 2;
  ring r = 0,(x,d),dp;
  ncalgebra(1,1); // Weyl-Algebra
  map F = r,x,-d;
  poly f =  x*d^2+d;
  poly If = involution(f,F);
  f-If;
  poly g = x^2*d+2*x*d+3*x+7*d;
  poly tg = -d*x^2-2*d*x+3*x-7*d;
  poly Ig = involution(g,F);
  tg-Ig;
  ideal I = f,g;
  ideal II = involution(I,F);
  II;
  I - involution(II,F);
  module M  = [f,g,0],[g,0,x^2*d];
  module IM = involution(M,F);
  print(IM);
  print(M - involution(IM,F));  
}
///////////////////////////////////////////////////////////////////////////////////
static proc new_var()
//generates a string of new variables
{

  int NVars=nvars(basering);
  int i,j;
  string s="@_1_1";
  for(i=1; i<=NVars; i++)
  {
    for(j=1; j<=NVars; j++)
    {
      if(i*j!=1)
      {
        s = s+ ","+NVAR(i,j);
      };    
    };
  };
  return(s);
};

static proc NVAR(int i, int j)
{
  return("@_"+string(i)+"_"+string(j));
};
///////////////////////////////////////////////////////////////////////////////////
static proc new_var_special()
//generates a string of new variables
{
  int NVars=nvars(basering);
  int i;
  string s="@_1_1";
  for(i=2; i<=NVars; i++)
  {
    s = s+ ","+NVAR(i,i);
  };
  return(s);
};
///////////////////////////////////////////////////////////////////////////////////
static proc RelMatr()
// returns the matrix of relations
// only Lie-type relations x_j x_i= x_i x_j + .. are taken into account
{
  int i,j;
  int NVars = nvars(basering);
  matrix Rel[NVars][NVars];
  for(i=1; i<NVars; i++)
  {
    for(j=i+1; j<=NVars; j++)
    {
      Rel[i,j]=var(j)*var(i)-var(i)*var(j);
    };
  };
  return(Rel); 
};
/////////////////////////////////////////////////////////////////
proc find_invo()
"USAGE: find_invo(); 
PURPOSE: describes a variety of linear involutions on a basering
RETURN: a ring together with a list of pairs L, where
@*        L[i][1]  =  Groebner Basis of an i-th associated prime,
@*        L[i][2]  =  matrix, defining a linear map, with entries, reduced with respect to L[i][1]
NOTE: for convenience, the full ideal of relations @code{idJ} 
and the initial matrix with indeterminates @code{matD} are exported in the output ring.
"{
  def @B    = basering; //save the name of basering
  int NVars = nvars(@B); //number of variables in basering
  int i, j;
  
  matrix Rel = RelMatr(); //the matrix of relations
   
  string s   = new_var(); //string of new variables
  string Par = parstr(@B); //string of parameters in old ring
  
  if (Par=="") // if there are no parameters
  { 
    execute("ring @@K=0,("+varstr(@B)+","+s+"), dp;"); //new ring with new variables
  }
  else //if there exist parameters
  {  
     execute("ring @@K=(0,"+Par+") ,("+varstr(@B)+","+s+"), dp;");//new ring with new variables
  };
  
  matrix Rel = imap(@B, Rel); //consider the matrix of relations in new ring
  
  int Sz = NVars*NVars+NVars; // number of variables in new ring
  
  matrix M[Sz][Sz]; //to be the matrix of relations in new ring
  
  for(i=1; i<NVars; i++) //initialize that matrix of relations
  {
    for(j=i+1; j<=NVars; j++)
    {
      M[i,j] = Rel[i,j];
    };
  };
  
  ncalgebra(1, M); //now new ring @@K become a noncommutative ring
  
  list l; //list to define an involution
  poly @@F;
  for(i=1; i<=NVars; i++) //initializing list for involution
  {
    @@F=0;
    for(j=1; j<=NVars; j++)
    {
      execute( "@@F = @@F+"+NVAR(i,j)+"*"+string( var(j) )+";" );
    };  
    l=l+list(@@F);  
  };
     
  matrix N = Rel; //imap(@B,Rel);
  
  for(i=1; i<NVars; i++)//get matrix by applying the involution to relations 
  {
    for(j=i+1; j<=NVars; j++)
    {
      N[i,j]= l[j]*l[i] - l[i]*l[j] + In_Poly( N[i,j], l, NVars);
    };
  };
  kill l;
  //---------------------------------------------
  //get the ideal of coefficients of N
  ideal J;  
  ideal idN = simplify(ideal(N),2);
  J = ideal(coeffs( idN, var(1) ) );
  for(i=2; i<=NVars; i++)
  {
    J = ideal( coeffs( J, var(i) ) ); 
  };
  J = simplify(J,2);
  //-------------------------------------------------
  if ( Par=="" ) //initializes the ring of relations 
  {
    execute("ring @@KK=0,("+s+"), dp;");
  }
  else
  {
    execute("ring @@KK=(0,"+Par+"),("+s+"), dp;");
  };
  ideal J = imap(@@K,J); // ideal, considered in @@KK now
  string snv = "["+string(NVars)+"]";
  execute("matrix @@D"+snv+snv+"="+s+";"); // matrix with entries=new variables
  
  J = J, ideal( @@D*@@D-matrix( freemodule(NVars) ) ); // add the condition that involution to square is just identity
  J = simplify(J,2); // without extra zeros
  list mL = minAssGTZ(J); // components not in GB
  int sL  = size(mL); 
  option(redSB);       // important for reduced GBs
  option(redTail);
  matrix IM = @@D;     // involution map
  list L    = list();  // the answer
  list TL;
  ideal tmp = 0;
  for (i=1; i<=sL; i++) // compute GBs of components
  {
    TL    = list();
    TL[1] = std(mL[i]);
    tmp   = NF( ideal(IM), TL[1] );
    TL[2] = matrix(tmp, NVars,NVars);
    L[i]  = TL;
  }
  export(L); // main export
  ideal idJ = J; // debug-comfortable exports
  matrix matD = @@D;
  export(idJ);
  export(matD);
  return(@@KK);
}
example
{ "EXAMPLE:"; echo = 2;
 def a = CreateWeyl(1);
 setring a; // this algebra is a first Weyl algebra
 def X = find_invo();
 setring X; // ring with new variables, corresponding to unknown coefficients
 L;
 print(L[1][2]);  // L[i][2] is a matrix in new variables, defining the linear involution
 L[1][1];  // where new variables obey these relations
}
///////////////////////////////////////////////////////////////////////////
proc find_invo_diag()
"USAGE: find_invo_diag();
PURPOSE: describes a variety of homothetic (diagonal) involutions on a basering
RETURN: a ring together with a list of pairs L, where
@*        L[i][1]  =  Groebner Basis of an i-th associated prime,
@*        L[i][2]  =  matrix, defining a linear map, with entries, reduced with respect to L[i][1]
NOTE: for convenience, the full ideal of relations @code{idJ} 
and the initial matrix with indeterminates @code{matD} are exported in the output ring.
"{
  def @B    = basering; //save the name of basering
  int NVars = nvars(@B); //number of variables in basering
  int i, j;
  
  matrix Rel = RelMatr(); //the matrix of relations
   
  string s   = new_var_special(); //string of new variables
  string Par = parstr(@B); //string of parameters in old ring
  
  if (Par=="") // if there are no parameters
  { 
    execute("ring @@K=0,("+varstr(@B)+","+s+"), dp;"); //new ring with new variables
  }
  else //if there exist parameters
  {  
    execute("ring @@K=(0,"+Par+") ,("+varstr(@B)+","+s+"), dp;");//new ring with new variables
  };
  
  matrix Rel = imap(@B, Rel); //consider the matrix of relations in new ring
  
  int Sz = 2*NVars; // number of variables in new ring
  
  matrix M[Sz][Sz]; //to be the matrix of relations in new ring
  for(i=1; i<NVars; i++) //initialize that matrix of relations
  {
    for(j=i+1; j<=NVars; j++)
    {
      M[i,j] = Rel[i,j];
    };
  };
  
  ncalgebra(1, M); //now new ring @@K become a noncommutative ring
  
  list l; //list to define an involution

  for(i=1; i<=NVars; i++) //initializing list for involution
  {
    execute( "l["+string(i)+"]="+NVAR(i,i)+"*"+string( var(i) )+";" );
  
  };
  matrix N = Rel; //imap(@B,Rel);
  
  for(i=1; i<NVars; i++)//get matrix by applying the involution to relations 
  {
    for(j=i+1; j<=NVars; j++)
    {
      N[i,j]= l[j]*l[i] - l[i]*l[j] + In_Poly( N[i,j], l, NVars);
    };
  };
  kill l;
  //---------------------------------------------
  //get the ideal of coefficients of N

  ideal J;  
  ideal idN = simplify(ideal(N),2);
  J = ideal(coeffs( idN, var(1) ) );
  for(i=2; i<=NVars; i++)
  {
    J = ideal( coeffs( J, var(i) ) ); 
  };
  J = simplify(J,2);
  //-------------------------------------------------
  
  if ( Par=="" ) //initializes the ring of relations 
  {
    execute("ring @@KK=0,("+s+"), dp;");
  }
  else
  {
    execute("ring @@KK=(0,"+Par+"),("+s+"), dp;");
  };
      
  ideal J = imap(@@K,J); // ideal, considered in @@KK now
  
  matrix @@D[NVars][NVars]; // matrix with entries=new variables to square i.e. @@D=@@D^2
  for(i=1;i<=NVars;i++)
  {
    execute("@@D["+string(i)+","+string(i)+"]="+NVAR(i,i)+";");
  };
  J = J, ideal( @@D*@@D - matrix( freemodule(NVars) ) ); // add the condition that involution to square is just identity
  J = simplify(J,2); // without extra zeros
  
  list mL = minAssGTZ(J); // components not in GB
  int sL  = size(mL); 
  option(redSB); // important for reduced GBs
  option(redTail);
  matrix IM = @@D; // involution map
  list L = list(); // the answer
  list TL; 
  ideal tmp = 0;
  for (i=1; i<=sL; i++) // compute GBs of components
  {
    TL    = list();
    TL[1] = std(mL[i]);
    tmp   = NF( ideal(IM), TL[1] );
    TL[2] = matrix(tmp, NVars,NVars);
    L[i]  = TL;
  }
  export(L);
  ideal idJ = J; // debug-comfortable exports
  matrix matD = @@D;
  export(idJ);
  export(matD);
  return(@@KK);
}
example
{ "EXAMPLE:"; echo = 2;
 def a = CreateWeyl(1);
 setring a; // this algebra is a first Weyl algebra
 def X = find_invo_diag();
 setring X; // ring with new variables, corresponding to unknown coefficients
 print(L[1][2]);  // a first matrix, defining the linear involution: we see it is constant
 print(L[2][2]);  // and a second possible matrix; it is constant too
 L; // let us take a look on the whole list
}
/////////////////////////////////////////////////////////////////////
proc find_auto()
"USAGE: find_auto();
PURPOSE: describes a variety of linear automorphisms of a basering
RETURN: a ring together with a list of pairs L, where
@*        L[i][1]  =  Groebner Basis of an i-th associated prime,
@*        L[i][2]  =  matrix, defining a linear map, with entries, reduced with respect to L[i][1]
NOTE: for convenience, the full ideal of relations @code{idJ} 
and the initial matrix with indeterminates @code{matD} are exported in the output ring.
"{
  def @B    = basering; //save the name of basering
  int NVars = nvars(@B); //number of variables in basering
  int i, j;
  
  matrix Rel = RelMatr(); //the matrix of relations
   
  string s   = new_var(); //string of new variables
  string Par = parstr(@B); //string of parameters in old ring
  
  if (Par=="") // if there are no parameters
  { 
    execute("ring @@K=0,("+varstr(@B)+","+s+"), dp;"); //new ring with new variables
  }
  else //if there exist parameters
  {  
     execute("ring @@K=(0,"+Par+") ,("+varstr(@B)+","+s+"), dp;");//new ring with new variables
  };
  
  matrix Rel = imap(@B, Rel); //consider the matrix of relations in new ring
  
  int Sz = NVars*NVars+NVars; // number of variables in new ring
  
  matrix M[Sz][Sz]; //to be the matrix of relations in new ring
  
  for(i=1; i<NVars; i++) //initialize that matrix of relations
  {
    for(j=i+1; j<=NVars; j++)
    {
      M[i,j] = Rel[i,j];
    };
  };
  
  ncalgebra(1, M); //now new ring @@K become a noncommutative ring
  
  list l; //list to define a homomorphism(isomorphism)
  poly @@F;
  for(i=1; i<=NVars; i++) //initializing list for involution
  {
    @@F=0;
    for(j=1; j<=NVars; j++)
    {
      execute( "@@F = @@F+"+NVAR(i,j)+"*"+string( var(j) )+";" );
    };  
    l=l+list(@@F);  
  };
     
  matrix N = Rel; //imap(@B,Rel);
  
  for(i=1; i<NVars; i++)//get matrix by applying the homomorphism  to relations 
  {
    for(j=i+1; j<=NVars; j++)
    {
      N[i,j]= l[j]*l[i] - l[i]*l[j] - Hom_Poly( N[i,j], l, NVars);
    };
  };
  kill l;
  //---------------------------------------------
  //get the ideal of coefficients of N
  ideal J;  
  ideal idN = simplify(ideal(N),2);
  J = ideal(coeffs( idN, var(1) ) );
  for(i=2; i<=NVars; i++)
  {
    J = ideal( coeffs( J, var(i) ) ); 
  };
  J = simplify(J,2);
  //-------------------------------------------------
  if ( Par=="" ) //initializes the ring of relations 
  {
    execute("ring @@KK=0,("+s+"), dp;");
  }
  else
  {
    execute("ring @@KK=(0,"+Par+"),("+s+"), dp;");
  };
  ideal J = imap(@@K,J); // ideal, considered in @@KK now
  string snv = "["+string(NVars)+"]";
  execute("matrix @@D"+snv+snv+"="+s+";"); // matrix with entries=new variables
  
  J = J, ideal( @@D*@@D-matrix( freemodule(NVars) ) ); // add the condition that homomorphism to square is just identity
  J       = simplify(J,2); // without extra zeros
  list mL = minAssGTZ(J); // components not in GB
  int sL  = size(mL); 
  option(redSB); // important for reduced GBs
  option(redTail);
  matrix IM = @@D; //  map
  list L = list(); // the answer
  list TL; 
  ideal tmp = 0;
  for (i=1; i<=sL; i++)// compute GBs of components
  {
    TL    = list();
    TL[1] = std(mL[i]);
    tmp   = NF( ideal(IM), TL[1] );
    TL[2] = matrix(tmp,NVars, NVars);
    L[i]  = TL;
  }
  export(L);
  ideal idJ = J; // debug-comfortable exports
  matrix matD = @@D;
  export(idJ);
  export(matD);
  return(@@KK);
}
example
{ "EXAMPLE:"; echo = 2;
 def a = CreateWeyl(1);
 setring a; // this algebra is a first Weyl algebra
 def X = find_auto();
 setring X; // ring with new variables, corresponding to unknown coefficients
 print(L[1][2]);  // a first matrix, defining the linear automorphism : we see it is constant
 print(L[2][2]);  // and a second possible matrix; it is constant too
 L; // let us take a look on the whole list
}