source: git/Singular/LIB/involut.lib @ e2ce0c

///////////////////////////////////////////////////////////////////
version="version involut.lib 4.1.2.0 Feb_2019 "; // $Id$
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";
LIB "ring.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(def 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
  {
    ring @@@KK = create_ring(0, "("+varstr(@B)+","+@ss+")", "dp"); //new ring with new variables
  }
  else //if there exist parameters
  {
     ring @@@KK = create_ring("(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
  {
    ring @@KK = create_ring(0, "("+@ss+")", "dp");
  }
  else
  {
    ring @@KK = create_ring("(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);
  intvec saveopt=option(get);
  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);
  option(set,saveopt);
  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
  {
    ring @@@KK = create_ring(0, "("+varstr(@B)+","+@ss+")", "dp"); //new ring with new variables 
  }
  else //if there exist parameters
  {
    ring @@@KK = create_ring("(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
  {
    ring @@KK = create_ring(0, "("+@ss+")", "dp"); 
  }
  else
  {
    ring @@KK = create_ring("(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);
  intvec saveopt=option(get);
  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);
  option(set,saveopt);
  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
  {
    ring @@@K = create_ring(0, "("+varstr(@B)+","+@ss+")", "dp");  //new ring with new variables
  }
  else //if there exist parameters
  {
     ring @@@K = create_ring("(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
  {
    ring @@KK = create_ring(0, "("+@ss+")", "dp"); 
  }
  if (( Par=="" ) && (n==0)) //initializes the ring of relations
  {
    ring @@KK = create_ring("(0,@p)", "("+@ss+")", "dp");
  }
  if ( Par!="" )
  {
    ring @@KK = create_ring("(0,"+Par+")", "("+@ss+")", "dp"); 
  }
  //  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);
  intvec saveopt=option(get);
  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);
  option(set,saveopt);
  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);
}