I am trying to compute the Hochschild cohomology of a particular bound quiver path algebra. The quiver $Q$ consists of one vertex and four loops $x,y, h_1,h_2$, and the relations $I$ are generated by:

1. All paths of length greater than 3.
2. All paths of length 3, except $yh_1x$ and $xh_2y$, and $yh_1x+xh_2y$.
3. All paths of length 2, except $yh_1, h_1x, xh_2,h_2y$.

Basically, in this algebra I have $yh_1x=-xh_2y$, and only other nonzero paths are the subpaths of these 2.

I am interested in $\mathit{HH}^2(kQ/I)$. More specifically, I am interested in whether $\mathit{HH}^2(kQ/I)=0$ for some infinite field $k$. I couldn't find or come up with a direct way of computing it, and my attempt using GAP's QPA package ran into memory problems. So I was wandering what are the tractable ways to compute this cohomology or prove that is zero or non-zero, either on paper or using computer algebra.

I am trying to compute the Hochschild cohomology of a particular bound quiver path algebra. The quiver $Q$ consists of one vertex and four loops $x,y, h_1,h_2$, and the relations $I$ are generated by:

1. All paths of length greater than 3.
2. All paths of length 3, except $yh_1x$ and $xh_2y$, and $yh_1x+xh_2y$.
3. All paths of length 2, except $yh_1, h_1x, xh_2,h_2y$.

Basically, in this algebra I have $yh_1x=-xh_2y$, and only other nonzero paths are the subpaths of these 2.

I am interested in $\mathit{HH}^2(kQ/I)$. More specifically, I am interested in whether $\mathit{HH}^2(kQ/I)=0$ for some infinite field $k$. I couldn't find or come up with a direct way of computing it, and my attempt using GAP's QPA package ran into memory problems. So I was wandering what are the tractable ways to compute this cohomology or prove that is zero or non-zero, either on paper or using computer algebra.

GAP code:

LoadPackage("qpa");
Q := Quiver(1, [[1,1,"x"],[1,1,"y"],[1,1,"h_1"],[1,1,"h_2"]]);
R := PathAlgebra(Rationals,Q);
gens:= GeneratorsOfAlgebra(R);
x:=gens[2];
y:=gens[3];
h_1:=gens[4];
h_2:=gens[5];
relations :=[x^2,y^2,h_1^2,h_2^2,xy,yx,h_1h_2,h_2h_1,xh_1x,xh_1y,yh_1y,xh_2x,yh_2x,yh_2y,yh_1x+xh_2y,h_1xh_1,h_1xh_2,h_2xh_1,h_2xh_2,h_1yh_1,h_1yh_2,h_2yh_1,h_2yh_2];
gb := GBNPGroebnerBasis(relations,R);
I:=Ideal(R,gb);
GroebnerBasis(I,gb);
A:=R/I;
M := AlgebraAsModuleOverEnvelopingAlgebra(A);
HH2 := ExtOverAlgebra(NthSyzygy(M, 1), M);

I am trying to compute the Hochschild cohomology of a particular bound quiver path algebra. The quiver $Q$ consists of one vertex and four loops $x,y, h_1,h_2$, and the relations $I$ are generated by:

1. All paths of length greater than 3.
2. All paths of length 3, except $yh_1x$ and $xh_2y$, and $yh_1x+xh_2y$.
3. All paths of length 2, except $yh_1, h_1x, xh_2,h_2y$.

Basically, in this algebra I have $yh_1x=-xh_2y$, and only other nonzero paths are the subpaths of these 2.

I am interested in $HH^2(kQ/I)$. More specifically, I am interested in whether $HH^2(kQ/I)=0$ for some infinite field $k$. I couldn't find or come up with a direct way of computing it, and my attempt using GAP's QPA package ran into memory problems. So I was wandering what are the tractable ways to compute this cohomology or prove that is zero or non-zero, either on paper or using computer algebra.

I am trying to compute the Hochschild cohomology of a particular bound quiver path algebra. The quiver $Q$ consists of one vertex and four loops $x,y, h_1,h_2$, and the relations $I$ are generated by:

1. All paths of length greater than 3.
2. All paths of length 3, except $yh_1x$ and $xh_2y$, and $yh_1x+xh_2y$.
3. All paths of length 2, except $yh_1, h_1x, xh_2,h_2y$.

Basically, in this algebra I have $yh_1x=-xh_2y$, and only other nonzero paths are the subpaths of these 2.

I am interested in $\mathit{HH}^2(kQ/I)$. More specifically, I am interested in whether $\mathit{HH}^2(kQ/I)=0$ for some infinite field $k$. I couldn't find or come up with a direct way of computing it, and my attempt using GAP's QPA package ran into memory problems. So I was wandering what are the tractable ways to compute this cohomology or prove that is zero or non-zero, either on paper or using computer algebra.