In his famous preprint Calabi-Yau algebras arXiv:math/0612139v3 [math.AG], Ginzburg seems to state that the Jacobian algebra of a cyclic quiver is Calabi-Yau algebra of dimension 3. There is a quote of the origin text:
Example 4.2.4 ......(omitted)
Example 4.2.5 (Cyclic quiver). Let Q be a quiver with $n+1$ vertices and edges $x_i\colon i\to i+1 \text{ mod } (n+1)$, which form an oriented cycle of length $n+1$. Let $\Phi$ be the cycle, $\Phi≔ x_1x_2\dotsc x_nx_{n+1}\text{ mod }[\mathbb{C}Q,\mathbb{C}Q]$. Then, $A=\mathfrak{A}(\mathbb{C}Q,\Phi)$ is a quotient of $\mathbb{C}Q$ by the two-sided ideal generated by all paths of length $\geq n$.
Remark 4.2.6. It is not difficult to show, using for instance Theorem 5.3.1 from §5.3 below, that the algebras A arising from the above examples are Calabi-Yau algebras of dimension 3.
If we take $n=2$ for simplicy, then $Q$ is a quiver with 3 vertices and 3 edges $a\colon 1\to 2, b\colon 2\to 3, c\colon 3\to 1$, and $I$ is the two-sided ideal generated by $ab, bc, ca$. The resulting quotient algebra $A=\mathbb{C}Q/I$ is a finite-dimensional algebra. However the above quote seems to claim that $A$ is a 3-Calabi-Yau algebra, which is be impossible since every finite-dimensional Calabi-Yau algebra is semisimple. Therefore I must have some misunderstandings of Ginzburg's lines.
P.S. I'm trying to calculate some easy examples of Calabi-Yau algebras arising from quivers with potential, and the cyclic quiver is my very first attempt. So any other hints on such kind of examples are also welcome!