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},{{{1,-1},{[b,c],[a,c]}},[a,b], {{1,-1},{[b,r4],[a,r4]}}}, genWeights => {{1,0},{1,0},{2,0},{3,1},{4,1},{4,2}}, genDiffs=>{[],[],[],{{-1},{[a,c]}}, [a,a,c],{{1,1},{[r4],[a,r3]}}},genSigns=>{0,0,0,1,1,0}) o1 = L o1 : LieAlgebra |
i2 : dimTableLie 5 o2 = | 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 | | 0 0 0 0 0 | 6 5 o2 : Matrix ZZ <--- ZZ |
i3 : eulerLie 5 o3 = {2, 1, 0, -1, -1} o3 : List |
i4 : homologyLie 5 o4 = | 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 o4 : Matrix ZZ <--- ZZ |