Changeset b9e6b3a in git

Timestamp:

Feb 21, 2005, 8:14:47 PM (18 years ago)

Author:

Viktor Levandovskyy <levandov@…>

Branches:

(u'spielwiese', '0d6b7fcd9813a1ca1ed4220cfa2b104b97a0a003')

Children:

34b03143f27c33c57d6458b8d0f9070e54067eed

Parents:

0e200a199387b736aed9e1893a5204ea814f2a66

Message:

*levandov: IntersectWithSub added, bug fixes, docu revisited and corrected


git-svn-id: file:///usr/local/Singular/svn/trunk@7739 2c84dea3-7e68-4137-9b89-c4e89433aadc

File:

• Singular/LIB/ncdecomp.lib (modified) (6 diffs)

Singular/LIB/ncdecomp.lib

@@ -1 +1 @@
 ///////////////////////////////////////////////////////////////////////////////
-version="$Id: ncdecomp.lib,v 1.4 2004-10-01 16:10:12 levandov Exp $";
+version="$Id: ncdecomp.lib,v 1.5 2005-02-21 19:14:47 levandov Exp $";
 category="Noncommutative";
 info="
@@ -6 +6 @@
 AUTHORS:  Viktor Levandovskyy,     levandov@mathematik.uni-kl.de.
 
+OVERVIEW:
+This library presents algorithms for the  central character 
+decomposition of a module (in other words, a 
+decomposition into generalized weight modules with respect to the center).
+Based on ideas of O. Khomenko and V. Levandovskyy.
+
 PROCEDURES:
-CentralQuot(I,G);       for a module I and an ideal G, returns I:G,
-CentralSaturation(M,T); for a module M and an ideal T, returns M:T^{\infty},
-CenCharDec(I,C);        for a module I and a list C of generators of a center, returns a list L, where each entry consists of three records, namely
-L[*][1] is the character, L[*][2] is the Groebner basis of corresponding weight module, L[*][3] is the K-dimension of the weight module;
+CentralQuot(I,G);       central quotient I:G,
+CentralSaturation(M,T); central saturation M:T:...:T ( = M:T^{\infty}),
+CenCharDec(I,C);        central character decomposition of I w.r.t. C
+IntersectWithSub(M,Z);  intersection of M with the subalgebra, generated
+by pairwise commutative elements.
 ";
 
@@ -132 +139 @@
 proc CentralSaturation(module M, ideal T)
 "USAGE:  CentralSaturation(M, T), for a module M and an ideal T,
-RETURN:  module of the central saturation M:T^{\infty},
+RETURN:  module of the central saturation of M by T (also denoted by M:T^{\infty}),
 NOTE:    the output module is not necessarily a Groebner basis,
 SEE ALSO: CentralQuot, CenCharDec
@@ -144 +151 @@
   while ( !MyIsEqual(Q,S) )
   {
-    Q = CentralQuot(M, T);
+    Q = CentralQuot(S, T);
     S = CentralQuot(Q, T);
   }
@@ -169 +176 @@
 }
 ///////////////////////////////////////////////////////////////////////////////
-proc CenCharDec(ideal I, list Center)
-"USAGE:  CenCharDec(I, L), I an ideal and Center a list of generators of the center;
-RETURN:  a list L, where each entry consists of three records, namely
-L[*][1] is the character, L[*][2] is the Groebner basis of corresponding weight module, L[*][3] is the K-dimension of the weight module;
-NOTE:    
+proc CenCharDec(module I, list L)
+"USAGE:  CenCharDec(I, L);  I a module, L a list of generators of the center;
+RETURN:  a list L, where each entry consists of three records:
+           L[*][1] is the character in terms of central elements, 
+           L[*][2] is the Groebner basis of the corresponding weight module, 
+           L[*][3] is the K-dimension of the weight module;
+NOTE:     some modules have no finite decomposition (in such a case one
+          gets warning message)
 SEE ALSO: CentralQuot, CentralSaturation
 EXAMPLE: example CenCharDec; shows examples
@@ -288 +298 @@
 { "EXAMPLE:"; echo = 2;
    option(returnSB);
-   def a = sl2();
+   def a = sl2(); // U(sl_2) in characteristic 0
    setring a;
    ideal I = e3,f3,h3-4*h;
-   I = twostd(I);
-   poly C=4*e*f+h^2-2*h;
-   list T = CenCharDec(I,C);
+   I = twostd(I);           // two-sided ideal generated by I
+   vdim(I);                 // it is finite-dimensional
+   list Cn = 4*e*f+h^2-2*h; // the only central element
+   list T = CenCharDec(I,Cn);
    T;
 }
+///////////////////////////////////////////////////////////////////////////////
+proc IntersectWithSub (ideal M, ideal Z)
+"USAGE:  IntersectWithSub(M,Z),  M an ideal, Z an ideal of pairwise commutative elements
+RETURN:  an intersection of M with the subalgebra, generated by Z
+NOTE:    usually one puts generators of the center into Z
+EXAMPLE: example IntersectWithSub; shows an example
+"
+{
+  // returns a submodule of M, equal to M \cap Z
+  // correctness: Z should consists of pairwise
+  // commutative elements
+  int nz = size(Z);
+  int i,j;
+  poly p;
+  for (i=1; i<nz; i++)
+  {
+    for (j=i+1; j<=nz; j++)
+    {
+      p = bracket(Z[i],Z[j]);
+      if (p!=0)
+      {
+        "Error: generators of the subalgebra do not commute.";
+        return(ideal(0));
+      }
+    }
+  }
+  // main action
+  def B = basering;
+  setring B;
+  string s1,s2;
+  s1 = "ring @Z = (";
+  s2 = s1 + charstr(basering) + "),(z(1.." + string(nz)+")),Dp";
+  //  s2;
+  execute(s2);
+  setring B;
+  map F = @Z,Z;
+  setring @Z;
+  ideal PreM = preimage(B,F,M);
+  PreM = std(PreM);
+  setring B;
+  ideal T = F(PreM);
+  return(T);
+}
+example
+{
+  "EXAMPLE:"; echo = 2;
+  ring r=(0,a),(e,f,h),Dp;
+  matrix @d[3][3];
+  @d[1,2]=-h;
+  @d[1,3]=2e;
+  @d[2,3]=-2f;
+  ncalgebra(1,@d); // parametric U(sl_2)
+  ideal I = e,h-a;
+  ideal C;
+  C[1] = h^2-2*h+4*e*f; // the center of U(sl_2)
+  ideal X = IntersectWithSub(I,C);
+  X;
+  ideal G = e*f, h; // the biggest comm. subalgebra of U(sl_2)
+  ideal Y = IntersectWithSub(I,G);
+  Y;
+}