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 : L=lieAlgebra({a,b,c,r3,r4,r42}, genWeights => {{1,0},{1,0},{2,0},{3,1},{4,1},{4,2}}, genSigns=>{0,0,0,1,1,0},diffl=>true) o1 = L o1 : LieAlgebra |
i2 : L.genDiffs={L.zz,L.zz,L.zz,a c,a a c,r4 - a r3} o2 = {0, 0, 0, (a c), (a a c), r4 - (a r3)} o2 : List |
i3 : Q=L/{b c - a c,a b,b r4 - a r4} o3 = Q o3 : LieAlgebra |
i4 : dimTableLie 5 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 |
i5 : eulerLie 5 o5 = {2, 1, 0, -1, -1} o5 : List |
i6 : homologyTableLie 5 o6 = | 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 o6 : Matrix ZZ <--- ZZ |