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VectorFields :: isLieAlgebra

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

Synopsis

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 SL2 on GL2 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 sl2, 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.

See also

Ways to use isLieAlgebra :