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

## Synopsis

• Usage:
b=isLieAlgebra(m)
• Inputs:
• m, , of vector fields
• Outputs:
• b, , 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.