commutator -- the commutator of a collection of vector fields

Synopsis

Usage:

N=commutator(M)

n=commutator(m)
Inputs:
- M, a matrix, of vector fields
- m, a module, of vector fields
Outputs:
- N, a matrix, with columns containing the brackets of pairs of columns of M
- n, a module, the module generated by brackets of elements of m

Description

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

Caveat

The Matrix and Module versions of this routine compute different things; see differences between certain bracketing functions.

Ways to use commutator :

commutator(Matrix)
commutator(Module)

For the programmer

The object commutator is a method function.