Tensor product of representations of quivers

Abstract

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 $Q^{\prime }$, and show that the representation $V\otimes W$ is semistable if $V$ and $W$ are semistable. We give a relation between the universal representations on the fine moduli spaces $N_1,N_2$ and $N_3$ of representations of $Q,Q^{\prime }$ and $Q\otimes Q^{\prime }$ respectively over arbitrary algebraically closed fields. We further describe a relation between the natural line bundles on these moduli spaces when the base is the field of complex numbers. We then prove that the internal product $\tilde{Q}\otimes \widetilde{Q^{\prime }}$ of covering quivers is a sub-quiver of the covering quiver $\widetilde{Q\otimes Q^{\prime }}$. We deduce the relation between stability of the representations $\widetilde{V\otimes W}$ and $\tilde{V}\otimes \tilde{W}$, where $\tilde{V}$ denotes the lift of the representation $V$ of $Q$ to the covering quiver $\tilde{Q}$. We also lift the relation between the natural line bundles on the product of moduli spaces $\widetilde{N_1}\times \widetilde{N_2}$.