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VectorFields :: isLieAlgebra

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



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


There is no isLieAlgebra(Matrix), yet.

See also

Ways to use isLieAlgebra :

For the programmer

The object isLieAlgebra is a method function.