If the Lie algebra has no differential, the program adds an extra last homological degree zero to each generator. The weight of [] is defined to be a list of zeroes of the same length as the weight of the generators. However, the weight of [] should be thought of as arbitrary since, in the example below, the element [b,b] has weight 2,2,0 and is equal to [] in L.
i1 : L=lieAlgebra({a,b,c},{},genWeights => {{1,1},{1,1},{2,2}}) o1 = L o1 : LieAlgebra |
i2 : weightLie(a) o2 = {1, 1, 0} o2 : List |
i3 : weightLie([a,a,b,a]) o3 = {4, 4, 0} o3 : List |
i4 : g={{1,-1},{[a,c],[b,c]}} o4 = {{1, -1}, {[a, c], [b, c]}} o4 : List |
i5 : weightLie g o5 = {3, 3, 0} o5 : List |
i6 : m=indexFormLie g o6 = mb - mb {3, 2} {3, 3} o6 : QQ[mb , mb , mb , mb , mb , mb , mb , mb ] {1, 0} {1, 1} {2, 0} {2, 1} {3, 0} {3, 1} {3, 2} {3, 3} |
i7 : degree m o7 = {3, 3, 0} o7 : List |
i8 : weightLie{[],[b,b],{{1,2},{[c],[a,b]}}} o8 = {{0, 0, 0}, {2, 2, 0}, {2, 2, 0}} o8 : List |
It is possible to use weightLie also in the case when the generators are indexedVariables or integers.
i9 : L2=lieAlgebra({a_2,b,1},{}, genWeights => {1,2,3}) o9 = L2 o9 : LieAlgebra |
i10 : weightLie(1) o10 = {3, 0} o10 : List |