On m-cluster tilted algebras and trivial extensions

The AG-invariant for (m+2)-angulations

“…Further, by the construction of (Q T , I T ), it immediately follows from the definition given in [15] that Λ is m-cluster-tilted of type A n . We note that recent work in [13] has shown how to construct m-cluster-tilted algebras from iterated tilted algebras with global dimension at most m + 1 via relation extensions, extending work that was done for m = 1 in [2]. The realization we have constructed above will only create m-cluster-tilted algebras with global dimension at most 2.…”

“…Quickly after, the notion of cluster-tilted algebras were introduced and studied in [1,2,6,7] to name only a few. This construction was then generalized to m-cluster categories and m-cluster-titled algebras in [16,14,13] among others. Roughly, the m-cluster category is defined as the orbit category D b (mod kQ)/ τ −1 [m] where Q is an acyclic quiver.…”

“…The m-cluster-tilted algebras are then defined as the endomorphism algebras of tilting objects in this category. For a complete description of this development we recommend [13,16]. We focus on the combinatorial description of m-cluster categories and the corresponding m-cluster-tilted algebras given via (m + 2)-angulations which has been studied in [5,9,15], we will generally adapt the definitions found in [15].…”