  | ### 7.3.11 lift (plural) `Syntax:` `lift (` ideal_expression`,` subideal_expression `)` `lift (` module_expression`,` submodule_expression `)` `Type:` matrix `Purpose:` computes the (left) transformation matrix which expresses the (left) generators of a submodule in terms of the (left) generators of a module. Uses different algorithms for modules which are (resp. are not) represented by Groebner bases. More precisely, if `m` is the module, `sm` the submodule, and `T` the transformation matrix returned by lift, then `transpose(matrix(sm)) = transpose(T)*transpose(matrix(m))`. If `m` and `sm` are ideals, `ideal(sm) = ideal(transpose(T)*transpose(matrix(m)))`. `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;
      def R=nc_algebra(1,D); // this algebra is a parametric U(sl_2)
      setring R;
      ideal I = e,h-a; // consider this parametric ideal
      I = std(I); // left Groebner basis
      print(matrix(I)); // print a compact presentation of I
      ==> h+(-a),e
      poly Z = 4*e*f+h^2-2*h; // a central element in R
      Z = Z - NF(Z,I); // a central character
      ideal j = std(Z);
      j;
      ==> j[1]=4*ef+h2-2*h+(-a2-2a)
      matrix T = lift(I,j);
      print(T);
      ==> h+(a+2),
      ==> 4*f
      ideal tj = ideal(transpose(T)*transpose(matrix(I)));
      size(ideal(matrix(j)-matrix(tj))); // test for 0
      ==> 0
See ideal (plural); liftstd (plural); module (plural).

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