# A-infinity Algebra Structures on Group Cohomology As described in [[Keller, 2001](../../references.md#cite-keller2001), [Keller, 2002](../../references.md#cite-keller2002)], an $A\infty$-algebra structure can be induced on $H^{*}A$ for any differential graded algebra $A$. Consider $\hbox{Ext}^{*}_R(S,S)$, 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 $\hbox{Ext}^{*}_R(S,S)$. An $A\infty$-algebra structure on a vector space $V$ consists of higher structural operations $m_1,\dots$ defined as $m_i:V^{\otimes i}\to V$ fulfilling the Stasheff axioms for all $n$: $$ \sum_{i+j-1=n,0\leq k\leq n-j}\pm m_i(a_1,\dots,a_{k-1},m_j(a_k,\dots,a_{k+j}),a_{k+j+1},\dots,a_n) = 0 $$ An $A\infty$-algebra homomorphism from an $A\infty$-algebra $A$ to an $A\infty$-algebra $B$ is a family $f_i$ of maps $A^{\otimes i}\to B$ such that the homomorphism axioms hold for all $n$: $\sum_{i+j-1=n,0\leq k\leq n-j} \pm f_i(a_1,\dots,a_{k-1},m_j(a_k,\dots,a_{k+j}),a_{k+j+1},\dots,a_n) =$$\sum_{i_1+\dots+i_r=n} \pm m_r(f_{i_1}(a_1,\dots,a_{i_1}),\dots,f_{i_r}(a_{n-i_r+1},\dots,a_n))$ According to a theorem fundamental to the algebraic uses of $A\infty$-techniques, for a differential graded algebra $A$, there is an $A\infty$-structure on $H^{*}A$ and an $A\infty$-algebra homomorphism $f:H^{*}A\to A$ such that $f_1$ 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 $\hbox{Ext}^{*}_{kG}(k,k)$ 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 $A\infty$-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 $A\infty$ object $Aoo$ corresponding to a group cohomology ring, this intrinsic calculates the structure map $m_i(t_1\otimes\dots\otimes t_i)$, where the $i$ give the length of terms, and $t_1,\dots,t_i$ are the elements of terms. ## `HighMap(Aoo, terms): Rec, SeqEnum[RngElt] -> MapChn` Given an $A\infty$ object $Aoo$ corresponding to a group cohomology ring, this intrinsic calculates the value of an $A\infty$-quasiisomorphism $f$ at the point $t_1\otimes\dots\otimes t_i$, where the $i$ gives the length of terms, and $t_1,\dots,t_i$ are the elements of terms. ## `Example: A-infinity mod 2 (ex-b55050)` The $A\infty$-structures on the cohomology rings of cyclic $p$-groups are well known examples in the literature: the $A\infty$-structure on $H^{*}(C_n,F_2)$ has one single higher structure nontrivial operation, namely $m_n$, which takes any $n$-tuple of odd coclasses to the even coclass of appropriate degree. In order to verify this for a specific example, we start by constructing an $A\infty$ record that contains all relevant information for the cohomology ring. ```magma > Aoo := AInfinityRecord(CyclicGroup(4),10); > S := Aoo`S; > HighProduct(Aoo,[x,x,x,x]); y > HighMap(Aoo,[x,x,x,x]); Basic algebra chain map of degree -1 ``` ## `Example: A-infinity mod 3 (ex-21e1bf)` The code as written handles odd characteristics well, with the sign choices featured in Kadeishvilis article [[Kadeishvili, 1980](../../references.md#cite-kadeishvili80)] embedded in the code. ```magma > Aoo := AInfinityRecord(CyclicGroup(3),10); > S := Aoo`S; > HighProduct(Aoo,[x,x,x]); y > HighMap(Aoo,[x,x,x]); Basic algebra chain map of degree -1 ``` ## Homological Algebra Toolkit For the computation of $A\infty$-structures, several methods are used that would invite a wider use in a generic homological algebra toolkit. ### `ActionMatrix(A, x): AlgBas, Mtrx -> ModMatFldElt` Produces a matrix of the right action of the Basic algebra element described by $x$ in the Basic algebra $A$. ### `CohomologyRingQuotient(CR): Rec -> Rng, Map` Computes the actual cohomology ring as a quotient ring of a multivariate polynomial ring from a cohomology ring record. ### `LiftToChainmap(P, f, d): ModCpx, Mtrx, RngIntElt -> MapChn` Lifts the function described by $f$ to a chain map from $P$ to $P$ of degree $d$. ### `NullHomotopy(f): MapChn -> MapChn` Constructs a null homotopy of the null homotopic chain map $f$. If $f$ is not null homotopic, the function will throw an error message, since in that case some of the equations encountered on the way are not solvable. ### `IsNullHomotopy(f, H): MapChn, MapChn -> BoolElt` Confirms that $H$ is a null homotopy of $f$, in other words that $f=dH-Hd$, with $d$ the differential of the corresponding chain complexes. ### `ChainmapToCohomology(f, CR): MapChn, Rec -> RngElt` Takes a chain map $f$ and returns the element in the cohomology quotient ring to which the chain map corresponds. ### `CohomologyToChainmap(xi, CR, P): RngElt, Rec, ModCpx -> MapChn` Takes an element $xi$ of a cohomology quotient ring of the cohomology ring record $CR$ and a projective resolution corresponding to that cohomology ring, and returns a chain map in the coclass represented by $xi$. ### `Example: Nullhomotopy (ex-ca2dae)` To illustrate the code that generates null homotopies, we consider the cohomology ring of a cyclic group, and pick out chain map representatives for the degree $1$ coclass. Although this squares to zero, the corresponding chainmaps do not compose to the zero chainmap. ```magma > A := BasicAlgebra(CyclicGroup(4)); > k := SimpleModule(A,1); > P := ProjectiveResolution(k,5); > R := CohomologyRing(k,5); > S := 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 ```