Blackbox computation of A-infinity structures
By: Mikael Vejdemo JohanssonIn my doctoral thesis (soon to be released to the general public), I spend some time discussing a blackbox technique for computing A-infinity structures on Ext algebras using Kadeishvili’s proof of the minimality theorem as an inspiration for an algorithmic approach.
Basically, we build up a quasi-isomorphism from to by only ever defining the bits and pieces we really need to define, and figuring out what they should be by staring at the Stasheff morphism axioms and recursing down until everything we need to know is computed.
During my visit to Sydney last autumn, I implemented this as a module for the computer algebra system MAGMA. Now, as a part of my last minute thesis revisions, I have tracked down and fixed (hopefully all) bugs in the computation. Hence, beginning with release 2.14-10 of MAGMA, there will be an A-infinity computation module distributed with the system which at least computes the already known A-infinity structures on correctly. It can also be used for explorations in p-group cohomology.
Some examples (the examples are run in MAGMA 2.14-9, but with a development implementation of the A-infinity module):
> G := CyclicGroup(3);
> Aoo := AInfinityRecord(G,10);
> S<x,y> := Aoo`S;
> HighProduct(Aoo,[ x : i in [1..10]]);
0
> { #k : k in Keys(Aoo`m) | not IsZero(HighProduct(Aoo,k)) };
{ 2, 3 }
> { #k : k in Keys(Aoo`m) | not IsZero(HighMap(Aoo,k)) };
{ 2 }
This example shows us that the only higher products and quasi-isomorphism components that do not vanish in have arity 2 and 3. With some extra verification, we can use some of the latest arguments in my thesis to convince ourselves that the entire A-infinity structure on this cohomology ring follows from this particular computation.
> G := DihedralGroup(4); > Aoo := AInfinityRecord(G,10); > S<x,y,z> := Aoo`S; > HighProduct(Aoo,[x,y,x,y]); z > HighProduct(Aoo,[x,y,x,y^2]); y*z > HighProduct(Aoo,[x,y,x,y^3]); y^2*z > HighProduct(Aoo,[x,y,x,y^4]); y^3*z
These are essentially a small, but representative subset of the computations on the cohomology of the dihedral group with 8 elements that I presented in T’bilisi in 2006.
So - if you have a MAGMA installation, 2.14-x, and want to play with this before 2.14-10 is released, prod me and we’ll work something out.
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Looks fun. I suppose it requires a group as input. Would it be conceivable to put in other sorts of augmented algebras instead, to find A-infinity structures on their Ext algebras?
(I’ve never used Magma, so I don’t know how much you’re using already existing Ext calculations, versus writing your own.)
Comment by John Palmieri — February 8, 2008 @ 9:38 pm
I’m using the Magma modules built by Jon F. Carlson for computing the cohomology data. Of course, once the homological algebra handling in Magma gets expanded to handle non-group-algebra rings, there’s nothing that prevents my code from working with those - however, the homological algebra code currently in there is built for the express purpose of computing group cohomology rings, and functions such as computing projective resolution and computing cohomology ring generators and relations only work if the BasicAlgebra (the data type for path algebra quotients) is a group ring.
Comment by Mikael Vejdemo Johansson — February 9, 2008 @ 2:08 am
Actually, it occurred to me. When computing Ext(k,k) over a group ring, we have the obvious way to construct, step by step, a minimal resolution. From the minimal resolution, we can deduce generators and relations. And we know when we have it all, as it were, by virtue of the Carlson/Benson stopping criteria.
For monomial rings, I know that there are good methods to get hold of the Poincare-Betti series given a presentation of the ring. And certainly, with an initial portion of a minimal resolution computed, it is easy to start constructing a finite presentation. However, I haven’t heard of anything resembling these stopping criteria.
John: are you able to sit down with a ring presentation (local, monomial, Adams graded - throw any nicety properties you want at it
and work out a finite presentation of Ext(k,k) as a k-algebra from it? Or is the obsession with Poincare-Betti series I remember rooted in ring structures being too difficult to get hold of?
Comment by Mikael Vejdemo Johansson — February 9, 2008 @ 1:50 pm
Mikael wrote:
This example shows us that the only higher products and quasi-isomorphism components that do not vanish in H *(C 3,Z/3) have arity 2 and 3.
Is it that arity greater than 3 is prohibited
by the order of the group?
e.g. would H *(C 5,Z/5) have arity up to 5?
Comment by Jim Stasheff — March 27, 2008 @ 6:00 pm
would indeed have arity up to 5, but it is a much more subtle interplay than just group order. For instance, my and Ainhoas results show that has unbounded structures with respect to the arity. So group order is far from the only factor in determining these structures.
Comment by Mikael Vejdemo Johansson — March 27, 2008 @ 7:35 pm