## Synopsis

• Usage:
M = ad(A,U,e)
• Inputs:
• A, an instance of the type DGAlgebra,
• U, , linear form in the generators of A
• e, an integer,
• Outputs:
• M, , with entries in the ground field

## Description

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