A Course on Cluster Tilted Algebras

Infinitesimal semi‐invariant pictures and co‐amalgamation

“…As 𝑀 𝜆 (𝜇𝑄) is nonsimple, no such morphism is an element of Hom [𝑘] (𝜇 𝑘 𝑀 𝜆 (𝑄), 𝜇 𝑘 𝑀 𝜆 (𝑄)). By Theorem 6.1.11 (4), this means…”

“…Suppose $$\Gamma$$ is an acyclic quiver and let $$Q = \mu _{k_t}\circ \cdots \circ \mu _{k_1}(\Gamma )$$ for some arbitrary sequence $$(k_1,\ldots ,k_t)$$. Then, by [9] (see also [4, section 2.2]), there exists an ideal $$I$$ so that the algebra $$KQ/I$$ is cluster‐tilted of type $$\Gamma$$, and conversely every cluster‐tilted algebra of type $$\Gamma$$ is realized in this way. The ideal $$I$$ is obtained by taking cyclic derivatives of a potential , which is a sum of cycles in the quiver $$Q$$, see, for example, [4, Proposition 2].…”

“…Then, by [9] (see also [4, section 2.2]), there exists an ideal $$I$$ so that the algebra $$KQ/I$$ is cluster‐tilted of type $$\Gamma$$, and conversely every cluster‐tilted algebra of type $$\Gamma$$ is realized in this way. The ideal $$I$$ is obtained by taking cyclic derivatives of a potential , which is a sum of cycles in the quiver $$Q$$, see, for example, [4, Proposition 2]. The generalization of mutation to quivers with potential is established in [18].…”

“…We consider tame cluster‐tilted algebras, which are precisely the endomorphism rings of clusters for tame hereditary algebras (see [24, section 3.4] or the introduction of [42]). It is shown in, for example, [4] that the Auslander–Reiten quiver of $$\Lambda$$ can be obtained by deleting the direct summands of $$M\sqcup P[1]$$ from the Auslander–Reiten quiver of $${\mathcal {C}}(H)$$. In particular, this implies that there will be a 1‐parameter family (parameterized by $$K^*$$) of homogeneous tubes in $$\mathsf {mod}\Lambda$$.…”