Given two path algebras $A$ and $B$, for example, A=: $1\to2$ B=: $1\to2\to3$, is the tensor product of A and B over a field $k$ a 
path algebra? if yes, how to represent it by a quiver? also, how to construct $(A,B)$-bimodules? 



The tensor product of two path algebras
It can be realized as a quotient of the Cartesian product of path algebras in question. You have to mod out by some commutativity relations.


