Tensor product of representations of quivers.

In this article, we define the tensor product V ⊗ W of a representation V of a quiver Q with a representation W of an another quiver Q ′ , and show that the representation V ⊗ 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 ′ and Q ⊗ Q ′ 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 Q ̃ ⊗ Q ′ ̃ of covering quivers is a sub-quiver of the covering quiver Q ⊗ Q ′ ˜. We deduce the relation between stability of the representations V ⊗ W ˜ and V ̃ ⊗ W ̃ , where V ̃ denotes the lift of the representation V of Q to the covering quiver Q ̃. We also lift the relation between the natural line bundles on the product of moduli spaces N 1 ̃ × N 2 ̃ . [ABSTRACT FROM AUTHOR]