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I am studyng this lemma of Assem:

enter image description here

Good, now let $Q$ the kronecker quiver :

enter image description here

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.

mathd
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J. Natch
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    Would you explain how the lemma gives you the isomorphism for the Kronecker quiver? I mean, the dimension of $KQ$ depends on $n$ but the one of the right hand side is constant $4$. Also the formats you wrote down do not really fit into a square matrix. Regarding your question, the $\varphi_{ik}^j$ are just restrictions of the multiplication map $KQ \otimes KQ \to KQ$. – Matthias Klupsch Jul 27 '17 at 06:28

1 Answers1

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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.

Pierre-Guy Plamondon
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  • Hi Pierre-Guy Plamondon, please a question, Assem says in an example about a path-algebra KQ that the defining basis is the set of all words on ${\alpha, \beta}$ with empty equal to $\varepsilon_1$. It refers to free monoids known as kleene star? – J. Natch Sep 02 '17 at 17:19