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7.5.3.0. isCartan
Procedure from librarycentral.lib (see central_lib).
- Usage:
- isCartan(f); f poly
- Purpose:
- check whether f is a Cartan element.
- Return:
- integer, 1 if f is a Cartan element and 0 otherwise.
- Note:
- f is a Cartan element of the algebra A
if and only if for all g in A there exists C in K such that [f, g] = C * g
if and only if for all variables v_i there exist C in K such that [f, v_i] = C * v_i.
LIB "central.lib"; ring R=0,(x,y,z),dp; matrix D[3][3]=0; D[1,2]=-z; D[1,3]=2*x; D[2,3]=-2*y; def r = nc_algebra(1,D); setring r; // this is U(sl_2) with cartan - z isCartan(z); // yes! ==> 1 poly p=4*x*y+z^2-2*z; isCartan(p); // central elements are Cartan elements! ==> 1 poly f=4*x*y; isCartan(f); // no way! ==> 0 isCartan( 10 + p + z ); // scalar + central + cartan ==> 1 |