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