isLieAlgebra -- check that a module of vector fields is closed under the Lie bracket

Synopsis

Usage:

b=isLieAlgebra(m)
Inputs:
- m, a module, of vector fields
Outputs:
- b, a Boolean value, whether the module is a Lie algebra of vector fields

Description

Checks whether the module generated by the provided vector fields is closed under the Lie bracket of vector fields (see bracket) and thus forms a Lie algebra.

i1 : R=QQ[a,b,c,d];

An action of SL_2 on GL_2 differentiates to the following vector fields:

i2 : e=matrix {{c},{d},{0},{0}};

             4       1
o2 : Matrix R  <--- R

i3 : f=matrix {{0},{0},{a},{b}};

             4       1
o3 : Matrix R  <--- R

i4 : h=matrix {{-a},{-b},{c},{d}};

             4       1
o4 : Matrix R  <--- R

Verify that this is sl_2, where [e,f]=h, [h,f]=-2f, [h,e]=2e.

i5 : bracket(e,f)-h==0

o5 = true

i6 : bracket(h,f)+2*f==0

o6 = true

i7 : bracket(h,e)-2*e==0

o7 = true

In particular, the module these generate form a Lie algebra:

i8 : isLieAlgebra(image (e|f|h))

o8 = true

Caveat

There is no isLieAlgebra(Matrix), yet.

Ways to use `isLieAlgebra` :

"isLieAlgebra(Module)"

For the programmer

The object isLieAlgebra is a method function.