Let $Q$ be a finite quiver. Then the following hold: (a) If $KQ$ is semisimple, then $|Q_1| = 0$.
If, moreover, $Q $ is connected, show that:
(b)$KQ$ is local only if $|Q_0| = 1$ and $|Q_1| = 0$,
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.Sign up to join this community
For (a) I guess it depends on the definition of semisimple. If you take Jacobson radical vanishes as the definition. Your claim is not true, since if you take the quiver with one vertex and a loop, you get the polynomial ring in one variable $K[X]$, whose Jacobson radical vanishes, but it has an arrow.
In Assem, Simson, Skowronski however, which seems to be the book you use (it is helpful if you include this data in your question), if I recall correctly, semisimple is only defined for finite-dimensional algebras. Then $KQ$ finite-dimensional implies that $Q$ has not oriented cycles. And in this case the ideal spanned by the arrows coincides with the Jacobson radical. Hence for the algebra to be semisimple there must not be arrows, hence $|Q_1|=0$.
For (b) if there are more than two vertices, then pick one of them, say $i$. Call the other $j$. Then $e_i$ is not invertible (since it is a zero divisor since $e_j\cdot e_i=0$). Similarly $1-e_i$ is not invertible. A contradiction to the property that for each local ring $x$ or $1-x$ is invertible. Thus $|Q_0|=1$. Now you are left with a polynomial ring (although with non-commuting variables). And you can use the excellent answers to your former question to conclude that there also must not be arrows.