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# lieSubSpace -- make a Lie subspace

- Usage:
    `S=lieSubSpace(gens)`
- Inputs:
  - gens, a list, a list of homogeneous elements in a Lie algebra $L$
- Outputs:
  - S, an instance of the type LieSubSpace, the subspace of $L$ generated by the list gens

## Description

The input should be a list of homogeneous Lie elements in a Lie algebra $L$. The output need not in general be invariant under the differential.

    i1 : F=lieAlgebra({a,b,c,r3,r4,r42},
            Weights => {{1,0},{1,0},{2,0},{3,1},{4,1},{4,2}},
            Signs=>{0,0,0,1,1,0},LastWeightHomological=>true)

    o1 = F

    o1 : LieAlgebra

    i2 : D=differentialLieAlgebra{0_F,0_F,0_F,a c,a a c,r4 - a r3}

    o2 = D

    o2 : LieAlgebra

    i3 : S=lieSubSpace{b c - a c,a b,b r4 - a r4}

    o3 = S

    o3 : LieSubSpace

    i4 : describe S

    o4 = generators => { - (a c) + (b c),  - (b a),  - (a r4) + (b r4)}
         lieAlgebra => D

    i5 : d=differential D

    o5 = d

    o5 : LieDerivation

    i6 : basis(5,S)

    o6 = {(a r4) - (b r4)}

    o6 : List

    i7 : d\oo

    o7 = {(a a a c) - (b a a c)}

    o7 : List

## See also

- lieAlgebra -- make a free Lie algebra
- lieSubAlgebra -- make a Lie subalgebra
- lieIdeal -- make a Lie ideal

## Ways to use lieSubSpace:

- `lieSubSpace(List)`

## For the programmer

The object lieSubSpace is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/GradedLieAlgebras/doc.m2:1149:0.