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3.9 lift

Syntax:
lift ( ideal_expression, subideal_expression )
lift ( module_expression, submodule_expression )
lift ( ideal_expression, subideal_expression, matrix_name )
lift ( module_expression, submodule_expression, matrix_name )
Type:
matrix
Purpose:
computes the left transformation matrix which expresses the generators of a submodule in terms of the generators of a module. Uses different algorithms for modules which are, resp. are not, represented by a standard basis.
More precisely, if m is the module (or ideal), sm the submodule (or ideal), and T the transformation matrix returned by lift, then transpose(matrix(sm)) = transpose(T)*transpose(m) and module(transpose(sm)) = module(transpose(T)*transpose(m)) (resp. ideal(sm) = ideal(transpose(T)*transpose(m))).
U is always the unity matrix if the basering is a polynomial ring (not power series ring). U is stored in the optional third argument.
Note:
Gives a warning if sm is not a submodule.
Example:
 
  ring r = (0,a),(e,f,h),(c,dp);
  matrix @D[3][3];
  @D[1,2]=-h;
  @D[1,3]=2*e;
  @D[2,3]=-2*f;
  ncalgebra(1,@D); // parametric U(sl_2)
  ideal i = e,h-a;
  i = std(i);
  i;
==> i[1]=h+(-a)
==> i[2]=e
  poly Z = 4*e*f+h^2-2*h; // central element
  Z = Z - NF(Z,i); //cen. character
  ideal j = std(Z);
  matrix T = lift(i,j);
  print(T);
==> h+(a+2),
==> 4*f     
  ideal tj = ideal(transpose(T)*transpose(matrix(i)));
  std(ideal(j-tj)); // test
==> _[1]=0
See ideal; liftstd; module.


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