A-infinity Algebra Structures on Group Cohomology

As described in [Kel01], [Kel02], an A∞-algebra structure can be induced on H ^* A for any differential graded algebra A. Consider Ext ^* _R(S, S), for R a quiver algebra quotient and S the direct sum of all simple R-modules. Regarded as the homology of the endomorphism algebra of a projective resolution of S, Keller further demonstrates how this additional algebraic structure allows recovery of R from Ext ^* _R(S, S).

An A∞-algebra structure on a vector space V consists of higher structural operations m₁, ... defined as m_i:V ^{tensor i}to V fulfilling the Stasheff axioms for all n:

∑_{i + j - 1=n, 0≤k≤n - j}∓ m_i(a₁, ..., a_{k - 1}, m_j(a_k, ..., a_{k + j}), a_{k + j + 1}, ..., a_n) = 0

An A∞-algebra homomorphism from an A∞-algebra A to an A∞-algebra B is a family f_i of maps A ^{tensor i}to B such that the homomorphism axioms hold for all n:

∑_{i + j - 1=n, 0≤k≤n - j} ∓ f_i(a₁, ..., a_{k - 1}, m_j(a_k, ..., a_{k + j}), a_{k + j + 1}, ..., a_n) = ∑_{i1 + ... + ir=n} ∓ m_r(f_i1(a₁, ..., a_i1), ..., f_ir(a_{n - ir + 1}, ..., a_n))

According to a theorem fundamental to the algebraic uses of A∞-techniques, for a differential graded algebra A, there is an A∞-structure on H ^* A and an A∞-algebra homomorphism f:H ^* A to A such that f₁ is a quasiisomorphism of differential graded algebras, and induced by the identity map on H ^* A.

A blackbox method of calculation can be based on Kadeishvilis' proof of this statement, using the homomorphism axioms to recursively calculate any specific values that are needed, and choosing the m_i and f_i in such a way as not to violate the axioms. The following package implements this method for the special case of Ext ^* _kG(k, k) for G a p-group, k a prime field of characteristic p and also the one-dimensional unique simple kG-module.

AInfinityRecord(G,n) : Grp, RngIntElt -> Rec

Constructs a record carrying all relevant information to calculate A∞-operations on a group cohomology ring. Among the data carried can be found the cohomology ring in R, the cohomology ring quotient in S, the projective resolution used in P, the simple module resolved in k and the basic algebra in A.

MasseyProduct(Aoo,terms) : Rec, SeqEnum[RngElt] -> RngElt

HighProduct(Aoo,terms) : Rec, SeqEnum[RngElt] -> RngElt

Given an A∞ object Aoo corresponding to a group cohomology ring, this intrinsic calculates the structure map m_i(t₁ tensor ... tensor t_i), where the i give the length of terms, and t₁, ..., t_i are the elements of terms.

HighMap(Aoo,terms) : Rec, SeqEnum[RngElt] -> MapChn

Given an A∞ object Aoo corresponding to a group cohomology ring, this intrinsic calculates the value of an A∞-quasiisomorphism f at the point t₁ tensor ... tensor t_i, where the i gives the length of terms, and t₁, ..., t_i are the elements of terms.

> Aoo := AInfinityRecord(CyclicGroup(4),10);
> S<x,y> := Aoo`S;
> HighProduct(Aoo,[x,x,x,x]);
y
> HighMap(Aoo,[x,x,x,x]);
Basic algebra chain map of degree -1

> Aoo := AInfinityRecord(CyclicGroup(3),10);
> S<x,y> := Aoo`S;
> HighProduct(Aoo,[x,x,x]);
y
> HighMap(Aoo,[x,x,x]);
Basic algebra chain map of degree -1

> A := BasicAlgebra(CyclicGroup(4));
> k := SimpleModule(A,1);
> P := ProjectiveResolution(k,5);
> R := CohomologyRing(k,5);
> S<x,y> := CohomologyRingQuotient(R);
> xi := CohomologyToChainmap(x,R,P);
> x*x;
0
> IsZero(xi*xi);
false
> ModuleMaps(xi*xi);
[*
    [0 0 1 0]
    [0 0 0 1]
    [0 0 0 0]
    [0 0 0 0],

    [0 0 1 0]
    [0 0 0 1]
    [0 0 0 0]
    [0 0 0 0],

    [0 0 1 0]
    [0 0 0 1]
    [0 0 0 0]
    [0 0 0 0],

    [0 0 1 0]
    [0 0 0 1]
    [0 0 0 0]
    [0 0 0 0]
*]
> H := NullHomotopy(xi*xi); 
> ModuleMaps(H);
[*
    [0 0 0 0]
    [0 0 0 0]
    [0 0 0 0]
    [0 0 0 0],

    [0 1 0 0]
    [0 0 1 0]
    [0 0 0 1]
    [0 0 0 0],

    [0 0 0 0]
    [0 0 0 0]
    [0 0 0 0]
    [0 0 0 0],

    [0 1 0 0]
    [0 0 1 0]
    [0 0 0 1]
    [0 0 0 0],

    [0 0 0 0]
    [0 0 0 0]
    [0 0 0 0]
    [0 0 0 0]
*]
> IsNullHomotopy(xi*xi,H);
true