I am studyng this lemma of Assem:
Good, now let $Q$ the kronecker quiver :
then there is algebra isomorphism
$KQ\cong \begin{bmatrix} K &0 \\ K^2 & K \end{bmatrix}$
where $K^2$ is considered as a K-K-bimódule in the way $a(x,y)=(ax,ay)$, $(x,y)b=(xb,yb)$ for all $a,b,x,y$ in $K$.
I would like to understand what it is $\varphi ^{j}_{ik}$ in the kronecker quiver, Or can it only be said that the K-linear map exists without being able to identify it? . Thanks.
As Matthias Klupsch has stated in a comment to the question, the map $\varphi_{ik}^j$ is the restriction of the multiplication map. Let us write it explicitly: $$ \array{ \varphi_{ik}^j: \varepsilon_i (KQ) \varepsilon_j \bigotimes_{\varepsilon_i (KQ) \varepsilon_i} \varepsilon_j (KQ) \varepsilon_k & \longrightarrow & \varepsilon_i (KQ) \varepsilon_k \\ \alpha \otimes \beta & \longmapsto & \alpha\beta. } $$ This map is well-defined and $K$-linear. To apply this to the Kronecker quiver, one simply has to specify the multiplication map.