# bracketMatrix -- Multiplication matrix of the homotopy Lie algebra

## Synopsis

• Usage:
M = bracketMatrix(A,d,e)
• Inputs:
• A, an instance of the type DGAlgebra, first part of the acyclic closure of a Koszul complex
• d, an integer,
• e, an integer,
• Outputs:
• M, , of linear forms in the generators of A

## Description

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