When given a Matrix, this returns a Matrix with columns containing the brackets of pairs of columns of M.
When given a Module, this returns the Module generated by $[X,Y]$, for $X$ and $Y$ in the module m. This is a superset of the module generated by the brackets of generators of m; see bracket for more details.
In either case, commutator computes much the same thing as calling bracket with repeated parameters, but is more efficient because it takes advantage of skew-symmetry.
See differences between certain bracketing functions for more information.
i1 : R=QQ[x,y]; |
i2 : D=derlog(ideal (x*y*(x-y)))
o2 = image | x 0 |
| y xy-y2 |
2
o2 : R-module, submodule of R
|
i3 : commutator(gens D)
o3 = | 0 |
| xy-y2 |
2 1
o3 : Matrix R <--- R
|
i4 : bracket(gens D,gens D)
o4 = | 0 0 0 0 |
| 0 xy-y2 -xy+y2 0 |
2 4
o4 : Matrix R <--- R
|
i5 : commutator(D)
o5 = image | 0 x2 0 0 0 xy 0 x2y-xy2 0 |
| xy-y2 xy x2y-xy2 0 0 y2 xy2-y3 xy2-y3 x2y2-2xy3+y4 |
2
o5 : R-module, submodule of R
|
i6 : bracket(D,D)
o6 = image | 0 0 0 0 x2 0 0 0 xy 0 x2y-xy2 0 x2 0 0 0 xy 0 x2y-xy2 0 |
| 0 xy-y2 -xy+y2 0 xy x2y-xy2 0 0 y2 xy2-y3 xy2-y3 x2y2-2xy3+y4 xy x2y-xy2 0 0 y2 xy2-y3 xy2-y3 x2y2-2xy3+y4 |
2
o6 : R-module, submodule of R
|
The Matrix and Module versions of this routine compute different things; see differences between certain bracketing functions.
The object commutator is a method function.