# 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

Briggs, Avramov

## Version

This documentation describes version 0.9 of HomotopyLieAlgebra.

## Source code

The source code from which this documentation is derived is in the file HomotopyLieAlgebra.m2.

## Exports

• Functions and commands
• allgens -- List the generators of a given degree
• bracket -- Computes the Lie product
• bracketMatrix -- Multiplication matrix of the homotopy Lie algebra
• Methods