This function implements the multiplication table of the degree d and degree e components of the homotopy Lie algebra Pi. The entries of the matrix are linear forms of homological degree d+e+1, interpreted as generators of Pi^{d+e}. See bracket for more details.
i1 : kk = ZZ/101 o1 = kk o1 : QuotientRing |
i2 : S = kk[x,y,z] o2 = S o2 : PolynomialRing |
i3 : R = S/ideal(x^2,y^2,z^2-x*y,x*z, y*z) o3 = R o3 : QuotientRing |
i4 : lastCyclesDegree = 4 o4 = 4 |
i5 : KR = koszulComplexDGA(ideal R)
o5 = {Ring => S }
Underlying algebra => S[T ..T ]
1 5
2 2 2
Differential => {x , y , - x*y + z , x*z, y*z}
o5 : DGAlgebra
|
i6 : A = acyclicClosure(KR, EndDegree => lastCyclesDegree); |
i7 : p1 = allgens(A,0) -- dual generators of Pi^1
o7 = {x, y, z}
o7 : List
|
i8 : p2 = allgens(A,1) -- dual generators of Pi^3
o8 = {T , T , T , T , T }
1 2 3 4 5
o8 : List
|
i9 : p3 = allgens(A,2) -- dual generators of Pi^4
o9 = {T , T , T , T , T }
6 7 8 9 10
o9 : List
|
i10 : bracketMatrix(A,2,1)
o10 = | 0 T_6 -T_7 |
| T_8 0 -T_9 |
| T_6 T_8 0 |
| T_7 -T_10 -T_6 |
| T_10 T_9 -T_8 |
5 3
o10 : Matrix (S[T ..T ]) <--- (S[T ..T ])
1 99 1 99
|
The object bracketMatrix is a method function.