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HomotopyLieAlgebra -- Homotopy Lie Algebra of a surjective ring homomorphism

Description

If R = S/I, K is the Koszul complex on the generators of I, and A is the DGAlgebra that is the acyclic closure of K, then the homotopy Lie algebra Pi of the map S -->> R is defined as in Briggs ****, with underlying vector space the graded dual of the space spanned by a given set of generators of A.

i1 : S = ZZ/101[x,y]

o1 = S

o1 : PolynomialRing
i2 : R = S/ideal(x^2,y^2,x*y)

o2 = R

o2 : QuotientRing
i3 : KR = koszulComplexDGA(ideal R)

o3 = {Ring => S                      }
      Underlying algebra => S[T ..T ]
                               1   3
                        2   2
      Differential => {x , y , x*y}

o3 : DGAlgebra

Since the acyclic closure is infinitely generated, we must specify the maximum homological degree in which cycles will be killed

i4 : lastCyclesDegree = 4

o4 = 4
i5 : A = acyclicClosure(KR, EndDegree => lastCyclesDegree)

o5 = {Ring => S                                                                                                                                                                                                                                                                                                                                                                                                                                                           }
      Underlying algebra => S[T ..T  ]
                               1   25
                        2   2                                                                                                                                                                                                                 2                                   2                                                                           2                                                                          2
      Differential => {x , y , x*y, x*T  - y*T , - y*T  + x*T , T T  + y*T , - T T  + x*T  + y*T , - T T  + x*T , - T T  + y*T , - T T  + x*T , - T T  - T T  + y*T , - T T  - T T  + x*T , - T T  + y*T , - T T  + x*T , - T T  + y*T , - 50T  - T T  + x*T , - T T  + y*T  , 50T  - T T  + y*T  , T T  + x*T   - y*T  , - T T  - T T  + x*T   + y*T  , - 50T  - T T  + x*T  , - T T  - T T  - T T  + y*T   + y*T  , - T T  + x*T  , 50T  - T T  + y*T  , - T T  + x*T  }
                                       2      3       1      3   2 3      4     1 2      4      5     1 3      5     2 4      6     3 4      6     3 4    2 5      7     1 4    3 5      7     3 5      8     1 5      8     2 6      9       4    3 6      9     3 6      10     4    2 7      11   4 5      11      12     1 6    3 7      10      12       5    1 7      12     4 5    3 7    2 8      12      13     3 8      13     5    3 8      14     1 8      14

o5 : DGAlgebra

The evaluation of bracketMatrix(A,d,e) gives the matrix of values of [Pi^d,Pi^e]. Here we are identifying the vector space spanned by the generators of A with its graded dual by taking the generators produced by the algorithm in the DGAlgebras package to be self-dual.

i6 : bracketMatrix(A,1,1)

o6 = | 2T_1 T_3  |
     | T_3  2T_2 |

                        2                 2
o6 : Matrix (S[T ..T  ])  <-- (S[T ..T  ])
                1   25            1   25
i7 : bracketMatrix(A,2,1)

o7 = | 0   -T_5 |
     | T_4 0    |
     | T_5 -T_4 |

                        3                 2
o7 : Matrix (S[T ..T  ])  <-- (S[T ..T  ])
                1   25            1   25
i8 : bracketMatrix(A,2,2)

o8 = | 0   -T_7 -T_8 |
     | T_7 0    T_6  |
     | T_8 -T_6 0    |

                        3                 3
o8 : Matrix (S[T ..T  ])  <-- (S[T ..T  ])
                1   25            1   25

Note that bracketMatrix(A,d,e) is antisymmetric in d,e if one of them is even, and symmetric in d,e if both are odd

i9 : bracketMatrix(A,1,1) - transpose bracketMatrix(A,1,1)

o9 = 0

                        2                 2
o9 : Matrix (S[T ..T  ])  <-- (S[T ..T  ])
                1   25            1   25
i10 : bracketMatrix(A,2,1) + transpose bracketMatrix(A,1,2)

o10 = 0

                         3                 2
o10 : Matrix (S[T ..T  ])  <-- (S[T ..T  ])
                 1   25            1   25

References

Briggs, Avramov

Author

Version

This documentation describes version 0.9 of HomotopyLieAlgebra, released October 19, 2021.

Citation

If you have used this package in your research, please cite it as follows:

@misc{HomotopyLieAlgebraSource,
  title = {{HomotopyLieAlgebra: A \emph{Macaulay2} package. Version~0.9}},
  author = {David Eisenbud},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}}
}

Exports

  • Functions and commands
    • ad -- matrix of the adjoint action
    • allgens -- List the generators of a given degree
    • bracket -- Computes the Lie product
    • bracketMatrix -- Multiplication matrix of the homotopy Lie algebra
  • Methods
    • ad(DGAlgebra,RingElement,ZZ) -- see ad -- matrix of the adjoint action
    • allgens(DGAlgebra) -- see allgens -- List the generators of a given degree
    • allgens(DGAlgebra,ZZ) -- see allgens -- List the generators of a given degree
    • bracket(DGAlgebra,List) -- see bracket -- Computes the Lie product
    • bracket(DGAlgebra,List,RingElement) -- see bracket -- Computes the Lie product
    • bracket(DGAlgebra,ZZ,ZZ) -- see bracket -- Computes the Lie product
    • bracketMatrix(DGAlgebra,ZZ,ZZ) -- see bracketMatrix -- Multiplication matrix of the homotopy Lie algebra

For the programmer

The object HomotopyLieAlgebra is a package, defined in HomotopyLieAlgebra.m2.


The source of this document is in /build/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/HomotopyLieAlgebra.m2:283:0.