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
|
Briggs, Avramov
This documentation describes version 0.9 of HomotopyLieAlgebra.
The source code from which this documentation is derived is in the file HomotopyLieAlgebra.m2.
The object HomotopyLieAlgebra is a package.