bracketMatrix -- Multiplication matrix of the homotopy Lie algebra

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, a matrix, 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   ]
                               1,1   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   }
       1,1   1,2   1,3   1,4   1,5

o8 : List

i9 : p3 = allgens(A,2) -- dual generators of Pi^4

o9 = {T   , T   , T   , T   , T   }
       2,1   2,2   2,3   2,4   2,5

o9 : List

i10 : bracketMatrix(A,2,1)

o10 = | 0       T_(2,1)  -T_(2,2) |
      | T_(2,3) 0        -T_(2,4) |
      | T_(2,1) T_(2,3)  0        |
      | T_(2,2) -T_(2,5) -T_(2,1) |
      | T_(2,5) T_(2,4)  -T_(2,3) |

                                                                   5                                                           3
o10 : Matrix (S[T   ..T   , T   ..T    , T   ..T    , T   ..T    ])  <-- (S[T   ..T   , T   ..T    , T   ..T    , T   ..T    ])
                 1,1   2,5   3,1   3,10   4,1   4,24   5,1   5,55            1,1   2,5   3,1   3,10   4,1   4,24   5,1   5,55

Ways to use bracketMatrix:

bracketMatrix(DGAlgebra,ZZ,ZZ)

For the programmer

The object bracketMatrix is a method function.

The source of this document is in /__w/M2/M2/macaulay2-1.26.05+ds/M2/Macaulay2/packages/HomotopyLieAlgebra.m2:417:0.

bracketMatrix -- Multiplication matrix of the homotopy Lie algebra

Description

See also

Ways to use bracketMatrix:

For the programmer