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A finite-dimensional $K$-algebra $A$ is of wild representation type if and only if there exists a $K\left<t_1,t_2\right>$-$A$-bimodule $_{K\left<t_1,t_2\right>}M_A$ such that the left $K\left<t_1,t_2\right>$-module $_{K\left<t_1,t_2\right>}M$ is finitely generated free and the induced functor $$-\otimes_{K\left<t_1,t_2\right>}M_A:\mathrm{fin}\,K\left<t_1,t_2\right>\rightarrow \mathrm{mod}\,A$$ respects isomorphism classes and maps indecomposable finite-dimensional $K\left<t_1,t_2\right>$-modules to indecomposable finitely generated $A$-modules.

The algebra $K\left<t_1,t_2\right>$ is an infinite-dimensional algebra which can be expressed as the path algebra of the quiver $Q$ with one vertex and two loops $t_1$ and $t_2$.

Let $X$ be a module $X \in \mathrm{fin}\,K\left<t_1,t_2\right>$, defined in terms of a quiver representation $(K^n, \varphi_{t_1}, \varphi_{t_2})$ of $Q$ (where $\varphi_{t_1}$ and $\varphi_{t_2}$ are linear maps $K^n \rightarrow K^n$).

Let $_{K\left<t_1,t_2\right>}M_{KQ'}$ be a $K\left<t_1,t_2\right>$-$KQ'$-bimodule, where $KQ'$ is a wild finite-dimensional path algebra. Suppose the right module $M_{KQ'}$, viewed as a representation of $Q'=(Q'_0,Q'_1)$, has the form $(M_i, \phi_a)_{i \in Q'_0, a \in Q'_1}$, where each vector space $M_i$ is a finite direct sum of copies of $K\left<t_1,t_2\right>$.

What does the right module $X\otimes_{K\left<t_1,t_2\right>}M_{KQ'}$ look like in terms of the quiver representation? Do we simply replace each copy of $K\left<t_1,t_2\right>$ with a copy of $K^n$ and replace $t_1$ (resp. $t_2$) with $\varphi_{t_1}$ (resp. $\varphi_{t_2}$) whenever $t_1$ (resp. $t_2$) is a matrix entry of a linear map $\phi_a$?

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  • $\begingroup$ I think you want the $\phi_a$ to be $K \langle t_1, t_2 \rangle$-morphisms. $\endgroup$ – Dag Oskar Madsen May 21 '17 at 8:42
  • $\begingroup$ It is certainly true that $\phi_a$ should be a $K\left<t_1, t_2\right>$-morphism, but $X \otimes_{K\left<t_1, t_2\right>} M_{KQ'}$ is a finitely generated (and finite-dimensional) right $A$-module. Can one describe the morphisms in the $K$-representation of $X \otimes_{K\left<t_1, t_2\right>} M_{KQ'}$? Specifically, if $X \otimes_{K\left<t_1, t_2\right>} M_{KQ'} = (N_i, \psi_a)_{i \in Q'_0,a \in Q'_1}$, what form does each $\psi_a$ take? Is it in any way related to the morphism $\phi_a$? $\endgroup$ – Iteraf May 22 '17 at 22:19
  • $\begingroup$ And for the vector spaces $N_i$, is this simply isomorphic to $(K^n)^m$ whenever $M_i$ is a direct sum of $m$ copies of $K\left<t_1, t_2\right>$? $\endgroup$ – Iteraf May 22 '17 at 22:20
  • $\begingroup$ It works as you describe, if you specify that the entries of the matrix $\phi_a$ should be elements of $K \langle t_1, t_2 \rangle$. I'll try to find a reference. $\endgroup$ – Dag Oskar Madsen May 24 '17 at 6:05

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