# lieSubSpace -- make a Lie subspace

## Synopsis

• 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

## Ways to use lieSubSpace :

• "lieSubSpace(List)"

## For the programmer

The object lieSubSpace is .