Changeset 6ef8674 in git for Singular/LIB

Timestamp:

Dec 3, 2010, 10:56:57 PM (13 years ago)

Author:

Viktor Levandovskyy <levandov@…>

Branches:

(u'spielwiese', 'fe61d9c35bf7c61f2b6cbf1b56e25e2f08d536cc')

Children:

847e4798004881faf06e4e596f53ce3bcb968bc2

Parents:

f940e2c0b05dd0f0cd990ecd5cde281ac39d25a1

Message:

*levandov: some new procs isLieType,isInvolution,isAntiEndo, cleanup, better doc, assumes

git-svn-id: file:///usr/local/Singular/svn/trunk@13725 2c84dea3-7e68-4137-9b89-c4e89433aadc

Location:

Singular/LIB

Files:

• involut.lib (modified) (9 diffs)
• nctools.lib (modified) (2 diffs)

Singular/LIB/involut.lib

@@ -8 +8 @@
 @*        Viktor Levandovskyy,  levandov@mathematik.uni-kl.de
 
-OVERVIEW:s
-Involution is an anti-isomorphism of a non-commutative K-algebra
+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
@@ -27 +27 @@
 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.
+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";
@@ -433 +436 @@
 @*        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
@@ -442 +447 @@
   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
@@ -564 +575 @@
 @*        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
@@ -572 +585 @@
   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
@@ -696 +715 @@
 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
@@ -709 +730 @@
     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
@@ -852 +881 @@
   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);
+};
+
+

Singular/LIB/nctools.lib

@@ -24 +24 @@
 findimAlgebra(M,[r]);     create finite dimensional algebra structure from the basering and the multiplication matrix M,
 superCommutative([b,e,Q]);  return qring, a super-commutative algebra over a basering,
-rightStd(I);              compute a right Groebner basis of an ideal,
-
+rightStd(I);              compute right Groebner basis of an ideal,
+rightNF(f,I);             compute right normal form wrt a submodule,
+rightModulo(M,N);         compute kernel of a homomorphism of right modules,
 moduloSlim(A,B);     compute modulo command via slimgb
 ncRelations(r);      recover the non-commutative relations of a G-algebra,
@@ -1779 +1780 @@
 }
 
+proc isLieType()
+"USAGE:  isLieType();
+RETURN:  int, 1 if basering is a G-algebra of Lie type, 0 otherwise
+PURPOSE: G-algebra of Lie type has relations of the kind Y*X=X*Y+D
+EXAMPLE: example isLieType; shows an example
+"
+{
+  def @B    = basering; //save the name of basering
+  int NVars = nvars(@B); //number of variables in basering
+  int i, j;
+
+  int answer = 1;  
+
+  // check basering is of Lie type:
+  matrix @@CC[NVars][NVars];
+  for(i=1; i<NVars; i++)
+  {
+    for(j=i+1; j<=NVars; j++)
+    {
+      @@CC[i,j]=leadcoef(var(j)*var(i));
+    };
+  };
+  ideal @C@ = simplify(ideal(@@CC),2+4);// skip zeroes and repeated entries
+  if (  (size(@C@) >1 ) || ( (size(@C@)==1) && (@C@[1]!=1) )  )
+  {
+    answer = 0;
+  };
+  return(answer);
+}
+example
+{
+  "EXAMPLE:"; echo = 2;
+  ring r = 0,(x,y),dp;
+  y*x;
+  isLieType(); //yes
+  def D = Weyl(); setring D;
+  y*x;
+  isLieType(); //yes
+  setring r;
+  def R = nc_algebra(-3,0); setring R;
+  y*x;
+  isLieType(); // no
+  kill R; kill r;
+  ring s = (0,q),(x,y),dp;
+  def S = nc_algebra(q,0); setring S;
+  y*x;
+  isLieType(); //no
+  kill S; setring s;
+  def S = nc_algebra(q,y^2); setring S;
+  y*x;
+  isLieType(); //no
+}