Cluster‐tilted algebras as trivial extensions

By Ibrahim Assem, Thomas Brustle, Ralf Schiffler. 2006

152 citations recorded by the source corpus.

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⋉ExtC2(DC,C) of C by the C–C‐bimodule ExtC2(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 C~ is cluster‐tilted if and only if there exists a tilted algebra C such that C~ is isomorphic to the relation‐extension of C.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b29d70edac9c0b34c3c456c6c934c84c24d580c4

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Citations in the corpus

Mutation of cluster-tilting objects and potentials (2008, methodology): “By [ ABS ] ([BR] in the Dynkin case) the quiver of EndCQ(T) is obtained from the quiver of the tilted algebra EndKQ′(T) by adding an arrow in the opposite direction for each relation in a minimal set of relations.”
Acyclic Calabi–Yau categories (2006, unknown): “This was used to prove significant new results on cluster algebras.”
On Cluster Algebras Arising from Unpunctured Surfaces (2007, methodology): “We recall some facts from cluster-tilting theory, see [BMRRT, BMR1, CCS, ABS , CK2] for details.”
Cluster-tilted algebras and slices (2007, background): “D is the Auslander-Reiten translation in D and [ 1 ] is the shift functor.”
Representations of Algebras and Related Topics (2011, background): “ Moreover the tensor product Ext2(DB,B)⊗B Ext2(DB,B) vanishes, hence the algebra Π3(B) is isomorphic to the trivial extension B ⊕Ext2(DB,B). The isomorphism Π3(B) ≃ EndCQ(T) recovers then a result of [11]. Note that in general, a τ2-ﬁnite algebra B of global dimension 2 can satisfy Ext2(DB,B) ⊗B Ext”
Cluster categories (2010, background): “The following result from [5] gives a procedure for passing from a tilted algebra to a cluster tilted algebra (see also ([114],[104])).”
Stable categories of higher preprojective algebras (2009, methodology): “This is a generalization of Crawley-Boevey’s description Λ̃ = TΛ Ext 1 Λ(DΛ,Λ) of classical preprojective algebras [CB1] and Assem-Brüstle-Schiffler’s description of cluster tilted algebras [ABS].”
Triangulated Categories: Cluster algebras, quiver representations and triangulated categories (2008, unknown): “ tensor product by T over the endomorphism algebra EndB(T) is an equivalence D(EndB(T)) →D∼ (B). The second assertion of part a) of the following theorem generalizes a result by Assem-Bru¨stleSchiﬄer [3]. Theorem 9.3 ([87]). a) The category C(Q,W) is triangle equivalent to the cluster category CA. T”
On the cluster category of a marked surface without punctures (2010, background): “Cluster categories associated to quivers are object of intense investigation (see for example [14, 8, 4, 5, 7, 15, 3]) and are understood explicitly in terms of representations of the quiver.”
Higher-dimensional cluster combinatorics and representation theory (2010, methodology): “ssion following Proposition 8.1 on why we use this deﬁnition. The following nice connection between tilting modules in addM and cluster tilting objects in OΛ remains true in this setup (see [BM+] and [ABS] for the triangulated case): 24 STEFFEN OPPERMANN AND HUGH THOMAS Theorem 5.5. Let T ∈addM be a”
Cluster-tilted algebras are Gorenstein and stably Calabi–Yau (2005, unknown): “W eshow thatthestablecategory ofitsCohen-M acaulay m odulesis3-CalabiYau.”