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Cluster-tilted algebras as trivial extensions  [PDF]
Ibrahim Assem,Thomas Brüstle,Ralf Schiffler
Mathematics , 2006,
Abstract: Given a finite dimensional algebra $C$ (over an algebraically closed field) of global dimension at most two, we define its relation-extension algebra to be the trivial extension $C\ltimes \Ext_C^2(DC,C)$ of $C$ by the $C$-$C$-bimodule $\Ext_C^2(DC,C)$. We give a construction for the quiver of the relation-extension algebra in case the quiver of $C$ has no oriented cycles. Our main result says that an algebra $\tilde C$ is cluster-tilted if and only if there exists a tilted algebra $C$ such that $\tilde C$ is isomorphic to the relation-extension of $C$.
Cluster-tilted algebras without clusters  [PDF]
Ibrahim Assem,Thomas Bruestle,Ralf Schiffler
Mathematics , 2009,
Abstract: Cluster-tilted algebras are trivial extensions of tilted algebras. This correspondence induces a surjective map from tilted algebras to cluster-tilted algebras. If B is a cluster-tilted algebra, we use the fibre of B under this map to study the module category of B. In particular, we introduce the notion of reflections of tilted algebras and define an algorithm that constructs the transjective component of the Auslander-Reiten quiver of cluster-tilted algebras of tree type.
On one-point extensions of cluster-tilted algebras  [PDF]
Miki Oryu,Ralf Schiffler
Mathematics , 2011,
Abstract: We define an operation which associates to a pair (B,M) where B is a cluster-tilted algebra and M is a B-module which lies in a local slice of B, a new cluster-tilted algebra B'. In terms of the quivers, this operation corresponds to adding one vertex (and arrows).
From iterated tilted algebras to cluster-tilted algebras  [PDF]
Michael Barot,Elsa Fernández,María Inés Platzeck,Nilda Isabel Pratti,Sonia Trepode
Mathematics , 2008,
Abstract: In this paper the relationship between iterated tilted algebras and cluster-tilted algebras and relation-extensions is studied. In the Dynkin case, it is shown that the relationship is very strong and combinatorial.
Hochschild cohomology of relation extension algebras  [PDF]
Ibrahim Assem,M. Andrea Gatica,Ralf Schiffler,Rachel Taillefer
Mathematics , 2015,
Abstract: Let $B$ be the split extension of a finite dimensional algebra $C$ by a $C$-$C$-bimodule $E$. We define a morphism of associative graded algebras $\varphi^*:\HH^*(B)\rightarrow \HH^*(C)$ from the Hochschild cohomology of $B$ to that of $C$, extending similar constructions for the first cohomology groups made and studied by Assem, Bustamante, Igusa, Redondo and Schiffler. In the case of a trivial extension $B=C\ltimes E$, we give necessary and sufficient conditions for each $\varphi^n$ to be surjective. We prove the surjectivity of $\varphi^1$ for a class of trivial extensions that includes relation extensions and hence cluster-tilted algebras. Finally, we study the kernel of $\varphi^1$ for any trivial extension, and give a more precise description of this kernel in the case of relation extensions.
Cluster-tilted algebras  [PDF]
Aslak Bakke Buan,Robert J. Marsh,Idun Reiten
Mathematics , 2004,
Abstract: We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation theory of hereditary algebras. As an application of this, we prove a generalised version of so-called APR-tilting.
Cluster-tilted algebras and slices  [PDF]
Ibrahim Assem,Thomas Bruestle,Ralf Schiffler
Mathematics , 2007,
Abstract: We give a criterion allowing to verify whether or not two tilted algebras have the same relation-extension (thus correspond to the same cluster-tilted algebra). This criterion is in terms of a combinatorial configuration in the Auslander-Reiten quiver of the cluster-tilted algebra, which we call local slice.
The algebras derived equivalent to gentle cluster tilted algebras  [PDF]
Grzegorz Bobinski,Aslak Bakke Buan
Mathematics , 2010,
Abstract: A cluster tilted algebra is known to be gentle if and only if it is cluster tilted of Dynkin type $\bbA$ or Euclidean type $\tilde{\bbA}$. We classify all finite dimensional algebras which are derived equivalent to gentle cluster tilted algebras.
Quivers with relations and cluster tilted algebras  [PDF]
Philippe Caldero,Frederic Chapoton,Ralf Schiffler
Mathematics , 2004,
Abstract: Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. To a cluster algebra of simply laced Dynkin type one can associate the cluster category. Any cluster of the cluster algebra corresponds to a tilting object in the cluster category. The cluster tilted algebra is the algebra of endomorphisms of that tilting object. Viewing the cluster tilted algebra as a path algebra of a quiver with relations, we prove in this paper that the quiver of the cluster tilted algebra is equal to the cluster diagram. We study also the relations. As an application of these results, we answer several conjectures on the connection between cluster algebras and quiver representations.
Constructing tilted algebras from cluster-tilted algebras  [PDF]
Marco Angel Bertani-?kland,Steffen Oppermann,Anette Wr?lsen
Mathematics , 2009, DOI: 10.1016/j.jalgebra.2010.02.026
Abstract: Any cluster-tilted algebra is the relation extension of a tilted algebra. We present a method to, given the distribution of a cluster-tilting object in the Auslander-Reiten quiver of the cluster category, construct all tilted algebras whose relation extension is the endomorphism ring of this cluster-tilting object.
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