//
version="$Id$";
category="Noncommutative";
info="
LIBRARY:  involut.lib  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

OVERVIEW:
Involution is an anti-automorphism of a non-commutative K-algebra
@* with the property that applied an involution twice, one gets an identity.
@* Involution is linear with respect to the ground field. In this library we compute
@* linear involutions, distinguishing the case of a diagonal matrix (such involutions
@* are called homothetic) and a general one. Also, linear automorphisms of different
@* order can be computed.

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

REMARK: This library provides algebraic tools for computations and operations
with algebraic involutions and linear automorphisms of non-commutative algebras

PROCEDURES:
findInvo();          computes linear involutions on a basering;
findInvoDiag();     computes homothetic (diagonal) involutions on a basering;
findAuto(n);          computes linear automorphisms of order n of a basering;
ncdetection();        computes an ideal, presenting an involution map on some particular noncommutative algebras;
involution(m,theta);  applies the involution to an object;
isInvolution(F); check whether a map F in an involution;
isAntiEndo(F);   check whether a map F in an antiendomorphism.
";

LIB "nctools.lib";
LIB "ncalg.lib";
LIB "poly.lib";
LIB "primdec.lib";
///////////////////////////////////////////////////////////////////////////////
proc ncdetection()
"USAGE:  ncdetection();
RETURN:  ideal, representing an involution map
PURPOSE: compute classical involutions (i.e. acting rather on operators than on variables) for some particular noncommutative algebras
ASSUME: the procedure is aimed at non-commutative algebras with differential, shift or advance operators arising in Control Theory.
It has to be executed in a ring.
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.
  if ( nameof(basering)=="basering" )
  {
    "No current ring defined.";
    return(ideal(0));
  }
  def r = basering;
  setring r;
  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;
  def r = nc_algebra(1,D); setring r;
  ncdetection();
  kill r, R;
  //----------------------------------------
  ring R=0,(x,S),dp;
  def r = nc_algebra(1,-S); setring r;
  ncdetection();
  kill r, R;
  //----------------------------------------
  ring R=0,(x,D(1),S),dp;
  matrix D[3][3];
  D[1,2]=1;  D[1,3]=-S;
  def r = nc_algebra(1,D); setring r;
  ncdetection();
}

static proc In_Poly(poly mm, list l, int NVars)
// applies the involution to the polynomial 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 polynomial 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
RETURN:  object of the same type as m
PURPOSE: applies the involution, presented by theta to the object m
THEORY: for an involution theta and two polynomials a,b from the algebra,
@*  theta(ab) = theta(b) theta(a); theta is linear with respect to the ground field
NOTE: This is generalized ''theta(m)'' for data types unsupported by ''map''.
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;
  def r = nc_algebra(1,1); setring r; // Weyl-Algebra
  map F = r,x,-d;
  F(F);  // should be maxideal(1) for an involution
  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;
  matrix(I) - involution(II,F);
  module M  = [f,g,0],[g,0,x^2*d];
  module IM = involution(M,F);
  print(IM);
  print(matrix(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";
  string s="a11";
  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));
  return("a"+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";
  string s="a11";
  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 findInvo()
"USAGE: findInvo();
RETURN: a ring containing a list L of pairs, where
@*        L[i][1]  =  ideal; a 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]
PURPOSE: computed the ideal of linear involutions of the basering
ASSUME: the relations on the algebra are of the form YX = XY + D, that is
the current ring is a G-algebra of Lie type.
NOTE: for convenience, the full ideal of relations @code{idJ}
and the initial matrix with indeterminates @code{matD} are exported in the output ring
SEE ALSO: findInvoDiag, involution
EXAMPLE: example findInvo; shows examples

"{
  def @B    = basering; //save the name of basering
  int NVars = nvars(@B); //number of variables in basering
  int i, j;

  // check basering is of Lie type:
  if (!isLieType())
  {
    ERROR("Assume violated: basering is of non-Lie type");
  }

  matrix Rel = RelMatr(); //the matrix of relations

  string @ss   = 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 @@@KK=0,("+varstr(@B)+","+@ss+"), dp;"); //new ring with new variables
  }
  else //if there exist parameters
  {
     execute("ring @@@KK=(0,"+Par+") ,("+varstr(@B)+","+@ss+"), 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];
    }
  }

  def @@K = nc_algebra(1, M); setring @@K; //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 = imap(@@@KK,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,("+@ss+"), dp;");
  }
  else
  {
    execute("ring @@KK=(0,"+Par+"),("+@ss+"), dp;");
  }
  ideal J = imap(@@K,J); // ideal, considered in @@KK now
  string snv = "["+string(NVars)+"]";
  execute("matrix @@D"+snv+snv+"="+@ss+";"); // 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 = makeWeyl(1);
  setring a; // this algebra is a first Weyl algebra
  a;
  def X = findInvo();
  setring X; // ring with new variables, corr. to unknown coefficients
  X;
  L;
  // look at the matrix in the new variables, defining the linear involution
  print(L[1][2]);
  L[1][1];  // where new variables obey these relations
  idJ;
}
///////////////////////////////////////////////////////////////////////////
proc findInvoDiag()
"USAGE: findInvoDiag();
RETURN: a ring together with a list of pairs L, where
@*        L[i][1]  =  ideal; a 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]
PURPOSE: compute homothetic (diagonal) involutions of the basering
ASSUME: the relations on the algebra are of the form YX = XY + D, that is
the current ring is a G-algebra of Lie type.
NOTE: for convenience, the full ideal of relations @code{idJ}
and the initial matrix with indeterminates @code{matD} are exported in the output ring
SEE ALSO: findInvo, involution
EXAMPLE: example findInvoDiag; shows examples
"{
  def @B    = basering; //save the name of basering
  int NVars = nvars(@B); //number of variables in basering
  int i, j;

  // check basering is of Lie type:
  if (!isLieType())
  {
    ERROR("Assume violated: basering is of non-Lie type");
  }

  matrix Rel = RelMatr(); //the matrix of relations

  string @ss   = 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 @@@KK=0,("+varstr(@B)+","+@ss+"), dp;"); //new ring with new variables
  }
  else //if there exist parameters
  {
    execute("ring @@@KK=(0,"+Par+") ,("+varstr(@B)+","+@ss+"), 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];
    }
  }

  def @@K = nc_algebra(1, M); setring @@K; //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 = imap(@@@KK,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,("+@ss+"), dp;");
  }
  else
  {
    execute("ring @@KK=(0,"+Par+"),("+@ss+"), 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 = makeWeyl(1);
  setring a; // this algebra is a first Weyl algebra
  a;
  def X = findInvoDiag();
  setring X; // ring with new variables, corresponding to unknown coefficients
  X;
  // print matrices, defining linear involutions
  print(L[1][2]);  // a first matrix: 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
  idJ;
}
/////////////////////////////////////////////////////////////////////
proc findAuto(int n)
"USAGE: findAuto(n); n an integer
RETURN: a ring together with a list of pairs L, where
@*        L[i][1]  =  ideal; a 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]
PURPOSE: compute the ideal of linear automorphisms of the basering,
@*  given by a matrix, n-th power of which gives identity (i.e. unipotent matrix)
ASSUME: the relations on the algebra are of the form YX = XY + D, that is
the current ring is a G-algebra of Lie type.
NOTE: if n=0, a matrix, defining an automorphism is not assumed to be unipotent
@* but just non-degenerate. A nonzero parameter @code{@@p} is introduced as the value of
@* the determinant of the matrix above.
@* For convenience, the full ideal of relations @code{idJ} and the initial matrix with indeterminates
@* @code{matD} are mutually exported in the output ring
SEE ALSO: findInvo
EXAMPLE: example findAuto; shows examples
"{
  if ((n<0 ) || (n==1))
  {
    "The index of unipotency is too small.";
    return(0);
  }


  def @B    = basering; //save the name of basering
  int NVars = nvars(@B); //number of variables in basering
  int i, j;

  // check basering is of Lie type:
  if (!isLieType())
  {
    ERROR("Assume violated: basering is of non-Lie type");
  }

  matrix Rel = RelMatr(); //the matrix of relations

  string @ss = 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)+","+@ss+"), dp;"); //new ring with new variables
  }
  else //if there exist parameters
  {
     execute("ring @@@K=(0,"+Par+") ,("+varstr(@B)+","+@ss+"), 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];
    }
  }

  def @@K = nc_algebra(1, M); setring @@K; //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 = imap(@@@K,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=="" ) && (n!=0)) //initializes the ring of relations
  {
    execute("ring @@KK=0,("+@ss+"), dp;");
  }
  if (( Par=="" ) && (n==0)) //initializes the ring of relations
  {
    execute("ring @@KK=(0,@p),("+@ss+"), dp;");
  }
  if ( Par!="" )
  {
    execute("ring @@KK=(0,"+Par+"),("+@ss+"), dp;");
  }
  //  execute("setring @@KK;");
  //  basering;
  ideal J = imap(@@K,J); // ideal, considered in @@KK now
  string snv = "["+string(NVars)+"]";
  execute("matrix @@D"+snv+snv+"="+@ss+";"); // matrix with entries=new variables

  if (n>=2)
  {
    J = J, ideal( @@D*@@D-matrix( freemodule(NVars) ) ); // add the condition that homomorphism to square is just identity
  }
  if (n==0)
  {
    J = J, det(@@D)-@p; // det of non-unipotent matrix is nonzero
  }
  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 = makeWeyl(1);
  setring a; // this algebra is a first Weyl algebra
  a;
  def X = findAuto(2);  // in contrast to findInvo look for automorphisms
  setring X; // ring with new variables - unknown coefficients
  X;
  size(L); // we have (size(L)) families in the answer
  // look at matrices, defining linear automorphisms:
  print(L[1][2]);  // a first one: we see it is the identity
  print(L[2][2]);  // and a second possible matrix; it is diagonal
  // L; // we can take a look on the whole list, too
  idJ;
  kill X; kill a;
  //----------- find all the linear automorphisms --------------------
  //----------- use the call findAuto(0)          --------------------
  ring R = 0,(x,s),dp;
  def r = nc_algebra(1,s); setring r; // the shift algebra
  s*x; // the only relation in the algebra is:
  def Y = findAuto(0);
  setring Y;
  size(L); // here, we have 1 parametrized family
  print(L[1][2]); // here, @p is a nonzero parameter
  det(L[1][2]-@p);  // check whether determinante is zero
}


proc isAntiEndo(def F)
"USAGE: isAntiEndo(F); F is a map from current ring to itself
RETURN: integer, 1 if F determines an antiendomorphism of
current ring and 0 otherwise
ASSUME: F is a map from current ring to itself
SEE ALSO: isInvolution, involution, findInvo
EXAMPLE: example isAntiEndo; shows examples
"
{
  // assumes:
  // (1) F is from br to br
  // I don't see how to check it; in case of error it will happen in the body
  // (2) do not assume: F is linear, F is bijective
  int n = nvars(basering);
  int i,j;
  poly pi,pj,q;
  int answer=1;
  ideal @f = ideal(F); list L=@f[1..ncols(@f)];
  for (i=1; i<n; i++)
  {
    for (j=i+1; j<=n; j++)
    {
      // F( x_j x_i) =def= F(x_i) F(x_j)
      pi = var(i);
      pj = var(j);
      //      q = involution(pj*pi,F) - F(pi)*F(pj);
      q = In_Poly(pj*pi,L,n) - F[i]*F[j];
      if (q!=0)
      {
        answer=0; return(answer);
      }
    }
  }
  return(answer);
}
example
{"EXAMPLE:";echo = 2;
  def A = makeUsl(2); setring A;
  map I = A,-e,-f,-h; //correct antiauto involution
  isAntiEndo(I);
  map J = A,3*e,1/3*f,-h; // antiauto but not involution
  isAntiEndo(J);
  map K = A,f,e,-h; // not antiendo
  isAntiEndo(K);
}


proc isInvolution(def F)
"USAGE: isInvolution(F); F is a map from current ring to itself
RETURN: integer, 1 if F determines an involution and 0 otherwise
THEORY: involution is an antiautomorphism of order 2
ASSUME: F is a map from current ring to itself
SEE ALSO: involution, findInvo, isAntiEndo
EXAMPLE: example isInvolution; shows examples
"
{
  // does not assume: F is an antiautomorphism, can be antiendo
  // allows to detect endos which are not autos
  // isInvolution == ( F isAntiEndo && F(F)==id )
  if (!isAntiEndo(F))
  {
    return(0);
  }
  //  def G = F(F);
  int j; poly p; ideal @f = ideal(F); list L=@f[1..ncols(@f)];
  int nv = nvars(basering);
  for(j=nv; j>=1; j--)
  {
    //    p = var(j); p = F(p); p = F(p) - var(j);
    //p = G(p) - p;
    p = In_Poly(var(j),L,nv);
    p = In_Poly(p,L,nv) -var(j) ;

    if (p!=0)
    {
      return(0);
    }
  }
  return(1);
}
example
{"EXAMPLE:";echo = 2;
  def A = makeUsl(2); setring A;
  map I = A,-e,-f,-h; //correct antiauto involution
  isInvolution(I);
  map J = A,3*e,1/3*f,-h; // antiauto but not involution
  isInvolution(J);
  map K = A,f,e,-h; // not antiauto
  isInvolution(K);
}