For each first degree $d$, where $d$ goes from $1$ to $n$, the alternating sum of the dimensions of the Lie algebra in homological degree 0 to $d-1$ is computed. As we know, the same numbers are obtained using the homology of the Lie algebra instead.
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
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i2 : L=differentialLieAlgebra{0_F,0_F,0_F,a c,a a c,r4 - a r3}
o2 = L
o2 : LieAlgebra
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i3 : Q=L/{b c - a c,a b,b r4 - a r4}
o3 = Q
o3 : LieAlgebra
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i4 : dims(5,Q)
o4 = | 2 1 1 1 2 |
| 0 0 1 3 5 |
| 0 0 0 1 2 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
5 5
o4 : Matrix ZZ <--- ZZ
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i5 : eulers(5,Q)
o5 = {2, 1, 0, -1, -1}
o5 : List
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i6 : H=lieHomology Q o6 = H o6 : VectorSpace |
i7 : dims(5,H)
o7 = | 2 1 0 0 0 |
| 0 0 0 1 1 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
5 5
o7 : Matrix ZZ <--- ZZ
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