In this article, we define the tensor product $V\otimes W$ of a representation $V$ of a quiver $Q$ with a representation $W$ of an another quiver $Q'$, and show that the representation $V\otimes W$ is semistable if $V$ and $W$ are semistable. Over the field of complex numbers, we also describe a relation between the natural line bundles, and between the universal representations on the fine moduli spaces $N_1, N_2$ and $N_3$ of representations of $Q, Q'$ and $Q\otimes Q'$ respectively. We then prove that the internal product $\tilde{ Q}\otimes \tilde{ Q'}$ of covering quivers is a sub-quiver of the covering quiver $\widetilde{ Q\otimes Q'}$. We deduce the relation between stability of the representations $\widetilde{ V\otimes W}$ and $\tilde{ V} \otimes \tilde{ W}$. We also lift the relation between natural line bundles on the product of moduli spaces $\tilde{ N_1} \times \tilde{ N_2}$.
Categories: math.AG