Computing Hochschild cohomology of an algebra in GAP

Question

I would like to calculate Hochschild cohomology of a path algebra of a quiver (modulo some relations) using GAP.

Creating the quiver and its path algebra modulo relations is explained in detail in the QPA manual.

After having defined an algebra $A$ I used

M := AlgebraAsModuleOverEnvelopingAlgebra(A);

to regard $A$ as a module over its enveloping algebra $A \otimes A^{\mathrm{op}}$.

There are a few steps left to calculate Hochschild cohomology:

1. find a projective resolution $P^\bullet$ of $M$,
2. take $\operatorname{Hom} (-, A)$ of the complex $P^\bullet$ (or rather its truncation?),
3. calculate the homology of the resulting complex.

For step 1, there are both ProjectiveResolution and ProjectiveResolutionOfPathAlgebraModule. Which would be preferable here?

Step 2 I don't know how to do in GAP.

(Step 3 should be HomologyOfComplex(C,i); where C is the output from step 2.)

Answer 1

In QPA one can compute $Ext$-groups via the command ExtOverAlgebra. Using the setup above with

M := AlgebraAsModuleOverEnvelopingAlgebra(A);

one can continue as follows:

HH1 := ExtOverAlgebra(M, M);
HH2 := ExtOverAlgebra(NthSyzygy(M, 1), M);
HH3 := ExtOverAlgebra(NthSyzygy(M, 2), M);
.......
HHn := ExtOverAlgebra(NthSyzygy(M, n - 1), M);

which computes the first, the second, the third, ..., the $n$-th Hochschild cohomology groups. The output from ExtOverAlgebra is a list of three elements, where the second entry is a list of basis vectors for the Ext-group as maps from the appropriate syzygy module to the extension module. If one is also interested in algebra structure, the command ExtAlgebraGenerators computes generators for the Ext-algebra (not necessarily minimal) up to a prescribed degree. For example,

ExtAlgebraGenerators(M, 4);

I hope that this is helpful.

Best regards, The QPA-team.