Tensor product of bimodules

Question

Im not sure whether this question is appropriate for MO, but I do not have much experience with bimodules (or I forgot many things).

Let $A$ be a finite dimensional (connected) algebra over a field $k$ and $M$ an indecomposable $A$-bimodule (finite dimensional prefered). For simplicity we can assume that $A$ is even a quiver algebra and maybe that the field is algebraically closed in case this is needed.

Is $M^{\otimes n}$ also indecomposable (or zero) for any $n \geq 1$ as an $A$-bimodule? If not, is it true under some extra conditions? What about when $M=D(A)$ ? (it seems to be this should be true at least for acyclic quiver algebras)

What can be said about the endomorphism ring of $M^{\otimes n}$ as an $A$-bimodule when the endomorphism ring of $M$ is known?

Here the tensor product is always over the algebra $A$. You can also give an answer for general rings $A$ and modules $M$ but I have no feeling whether this question is interesting or trivial in such generality.

Answer 1

Let $Q=( 2_{\circlearrowleft c}\xleftarrow a 1 \xrightarrow b 3_{\circlearrowleft d})$, that is, 3 vertices with 4 arrows, one loop at vertex 2 and 3 and two arrows from vertex 1 to 2 and 3. Consider the relations $\{ac, c^2, d^2, bd\}$ on $Q$. Define $A = \mathbb{Z}_7Q/\langle ac, c^2, d^2, bd\rangle$.

Let $D(A)$ be the dual of $A$ as a bimodule over $A$. Then given that QPA2 doesn't do any mistakes, $DM^{\otimes 2}$ is decomposable. Here are the computations done in QPA2:

gap> Q := RightQuiver( "Q(3)[a:1->2,b:1->3,c:2->2,d:3->3]" );       
Q(3)[a:1->2,b:1->3,c:2->2,d:3->3]
gap> KQ := PathAlgebra(GF(7), Q);
GF(7) * Q
gap> rels := [ KQ.ac,KQ.cc,KQ.dd,KQ.bd ]; 
[ Z(7)^0*(a*c), Z(7)^0*(c*c), Z(7)^0*(d*d), Z(7)^0*(b*d) ]
gap> A := KQ/rels;                        
(GF(7) * Q) / [ Z(7)^0*(a*c), Z(7)^0*(c*c), Z(7)^0*(d*d), Z(7)^0*(b*d) ]
gap> M := AlgebraAsBimodule(A);           
<1,1,1,0,2,0,0,0,2>
gap> DM := DualOfModule(M);               
<1,1,1,0,2,0,0,0,2>
gap> DM2 := TensorProductOfModules(DM,DM);
<0,1,1,0,1,0,0,0,1>
gap> DM3 := TensorProductOfModules(DM,DM2);
<0,1,1,0,1,0,0,0,1>
gap> IsIndecomposableModule(DM2);
false
gap> DecomposeModule(DM2);
[ <0,0,0,0,0,0,0,0,1>, <0,0,0,0,1,0,0,0,0>, <0,0,1,0,0,0,0,0,0>, <0,1,0,0,0,0,0,0,0> ]
gap> IsomorphicModules(DM2,DM3);
true

Here QPA2 claims that $D(A)^{\otimes 2}$ decomposes in 4 simple bimodules. Furthermore, the bimodules $D(A)^{\otimes n}$ are all isomorphic for $n\geq 2$, and $\operatorname{End}(D(A)^{\otimes 2}) \simeq \mathbb{Z}_7^4$.

I hope that these comments are helpful.

The QPA-team.