# differentialLieAlgebra -- make a differential Lie algebra

## Synopsis

• Usage:
L=differentialLieAlgebra(defs)
• Inputs:
• defs, a list, of Lie elements in a free Lie algebra
• Outputs:

## Description

The input should be a list of Lie elements in a free Lie algebra $F$ and this list consists of the differentials of the generators, where $0_F$ is used for the zero element. The option LastWeightHomological for $F$ must have the value true. The program adds relations to the Lie algebra to get the square of the differential to be 0. Use ideal(LieAlgebra) to get these non-normalized relations or describe(LieAlgebra) to just look at them.

 i1 : F1=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 = F1 o1 : LieAlgebra i2 : D1=differentialLieAlgebra{0_F1,0_F1,0_F1,a c,a a c,r4 - a r3} o2 = D1 o2 : LieAlgebra i3 : ideal D1 o3 = {(a a c) - (a a c)} o3 : List i4 : F2=lieAlgebra({a,b,c2,c3,c5},Signs=>{0,0,1,0,1}, Weights=>{{1,0},{1,0},{2,1},{3,2},{5,3}}, LastWeightHomological=>true) o4 = F2 o4 : LieAlgebra i5 : D2=differentialLieAlgebra{0_F2,0_F2,a b,a c2,a b c3} o5 = D2 o5 : LieAlgebra i6 : describe D2 o6 = generators => {a, b, c2, c3, c5} Weights => {{1, 0}, {1, 0}, {2, 1}, {3, 2}, {5, 3}} Signs => {0, 0, 1, 0, 1} ideal => { - (a b a), (a b a c2)} ambient => F2 diff => {0, 0, - (b a), (a c2), (a b c3)} Field => QQ computedDegree => 0