INMABB   05456
INSTITUTO DE MATEMATICA BAHIA BLANCA
Unidad Ejecutora - UE
artículos
Título:
Trivial extensions, iterated tilted algebras and cluster tilted algebras.
Autor/es:
MARÍA INÉS PLATZECK
Revista:
Sao Paulo Journal of Mathematical Sciences
Editorial:
IME- Universidade de Sao Paulo0
Referencias:
Lugar: Sao Paulo; Año: 2009
ISSN:
1982-6907
Resumen:
Classical results in representation theory establish very interesting connections between iterated tilted algebras and trivial extensions of finite dimensional algebras, particularly deep and useful in the Dynkin case. Recent results show that there are also interesting connections betweeen iterated tilted algebras and cluster tilted algebras. The aim of this article is to describe these connections, and show that, though the situations are quite different there are strong analogies between them. The relation between the properties of the trivial extension of a finite dimensional algebra Λ and those of the algebra itself have been the object of study by many mathematicians. Early work in this direction was done by H. Tachikawa (1980) who proved that the hereditary algebra Λ is of finite representation type if and only if the trivial extension T(Λ) of Λ is of finite representation type. On the other hand, K. Yamagata proved when Λ has oriented cycles then T(Λ) is of infinite representation type (1981). The connections with tilting theory are given by two theorems, due to Hughes and Waschb¨usch (1983) and to Assem, Happel and Rold´an (1984), respectively. The first proves that the trivial extension of Λ is of finite representation type and Cartan class Q if and only if there exists a tilted algebra Λ0 of Dynkin type Q such that T(Λ) ´ T(Λ0). The second establishes that T(Λ) is of finite representation type and Cartan class Q if and only if there exists a tilted algebra Λ0 of Dynkin type Q such that T(Λ) ´ T(Λ0). A combinatorial method to decide wether two schurian algebras Λ and Λ0 have isomorphic trivial extensions was given by Fern´andez in [23] (1999) using the notion of admissible cut. A subset Δ of the set of arrows of a quiver Q is called an admissible cut if each oriented minimal (or chordless) cycle of Q contains exactly one arrow of Δ. A quotient by an admissible cut of an algebra Λ is defined as the quotient of Λ by the ideal generated in Λ by an admissible cut in the quiver of Λ. It is proven in [23] that two schurian algebras Λ and Λ0 have isomorphic trivial extensions if and only if Λ0 is the quotient of T(Λ) by an admissible cut. Thus iterated tilted algebras of Dynkin type coincide with quotients of trivial extensions of finite representation type by admissible cuts. Combining these results Fern´andez classified all trivial extensions of finite representation type, giving a simple method to decide if an algebra is iterated tilted 2000 Mathematics Subject Classification. Primary: 16G20, Secondary: 16G70, 18E30. The author is researcher from CONICET, Argentina and thankfully acknowledges partial support from CONICET and from Universidad Nacional del Sur, Argentina. 1 2 PLATZECK of a given Dynkin type. Moreover, she obtained, under a unified approach, the classification results for iterated tilted algebras of types An, Dn and E6 obtained by different auhors with other methods ([4], [10], [34], [43]). In connection with cluster algebras, defined and studied by Fomin and Zelevinski in 2000, cluster categories were defined by Buan, Marsch, Reinecke, Reiten and Todorov and a tilting theory was developed for them [16]. To each hereditary algebra a cluster algebra can be associated, in such a way that cluster variables correspond to indecomposable rigid (or exceptional) objects and clusters to cluster tilting objects in the cluster category of H. Since then, the theory has had an extraordinary developement in different directions, with interesting connections to several areas of mathematics. We are interested here in the relation between cluster tilted algebras, relation extensions and iterated tilted algebras. Cluster tilted algebras are defined as endomorphism rings of cluster tilting objects in the cluster category of a hereditary algebra H. The connection of cluster tilted algebras with tilted algebras was studied by Assem, Br¨ustle and Schiffler, and is given using the notion of relation extension. Given an algebra Λ of global dimension at most two, the relation extension of Λ is the trivial extension R(Λ) = Λ Ext2 Λ(DΛ,Λ). They prove that an algebra is cluster tilted if and only if it is ismorphic to the relation extension of a tilted algebra. This result resembles the first of the results connecting tilting theory and trivial extensions above mentioned, but it is in some sense more general, since no assumption about representation type is made. As for trivial extensions, there is also a connection between cluster tilted algebras and iterated tilted algebras, but in this case only with those of global dimension at most two [1, 11, 31]. Given an iterated tilted algebra B, then B = EndDb(H)(T), where Db(H) denotes the derived category of a hereditary algebra H and T is a tilting complex in Db(H). When gldimB ≤ 2, then T defines a cluster tilting object in the cluster category C(H) and C = EndC(T) is a cluster tilted algebra. Moreover, there exists a sequence of algebra homomorphisms B → C π −→ R(B) → B whose composition is the identity of B and Ker(π) ⊆ rad2 C. In particular, C and R(B) have the same quivers. In contrast with the situation for trivial extensions, it is not required here that B is of Dynkin type. However, it is not true in general that C ´ R(B), not even in the Dynkin case. Finally, we turn our attention to admissible cuts of cluster tilted algebras of finite representation type, where the following result, proven in [11], holds: An algebra B with gldimB ≤ 2 is iterated tilted of Dynkin type Q if and only if it is the quotient of a cluster-tilted algebra of type Q by an admissible cut. These results can be applied to classify cluster tilted algebras of finite type. In fact, combining them Bordino, Fern´andez and Trepode classified those of type Ep, as communicated by Fern´andez in ICRA XII, 2008.