The adjoint action of a scalar linear combination of the entries of allgens(A,d-1) U, regarded as an element of Pi^d, acts by bracket multiplcation with source Pi^e and target Pi^{d+e}. The output is a matrix whose columns correspond to a generalized row of the output of bracketMatrix. bracketmatrix
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 : d = 1 o7 = 1 |
i8 : e = 1 o8 = 1 |
i9 : U = sum (Gd = allgens(A,d-1))
o9 = x + y + z
o9 : S[T ..T ]
1 99
|
i10 : ad(A,U,1)
o10 = {1, 2} | 2 0 0 |
{1, 2} | 0 2 0 |
{1, 2} | -1 -1 2 |
{1, 2} | 1 0 1 |
{1, 2} | 0 1 1 |
5 3
o10 : Matrix (S[T ..T ]) <--- (S[T ..T ])
1 99 1 99
|
The columns of this matrix are the functionals that are the sum of the three rows of the bracket multiplication table:
i11 : matrix{{1,1,1}}*bracketMatrix(A,d,e)
o11 = | 2T_1-T_3+T_4 2T_2-T_3+T_5 2T_3+T_4+T_5 |
1 3
o11 : Matrix (S[T ..T ]) <--- (S[T ..T ])
1 99 1 99
|
The object ad is a method function.