Dynkin Cluster-tilted Algebras And M-cluster-tilted Algebras

In 2002, cluster algebras were introduced by Fomin and Zelevinsky [FZ1, FZ2] in order to explain the connection between the canonical basis of a quantized enveloping algebra and total positivity for algebraic groups. There are interesting connections to their theory in many directions (see [FZ3]). As a categorical model for better understanding of cluster algebras, cluster categories and cluster-tilting theory were introduced in [BMRRT]. Let H be a finite dimensional hereditary algebra over an algebraically closed field and Db(H) be the bounded derived category of H. Then the cluster category (?)(H) is the orbit category Db(H)/τ-1[1], whereτis the Auslander-Reiten translation in Db(H) and [1] is the shift functor of Db(H). There is a one to one correspondence between cluster tilting objects in cluster categories and the clusters of cluster algebras.Cluster-tilted algebras were introduced in [BMR1], which together with cluster categories provide an algebraic understanding of combinatorics of cluster algebras. Here, cluster-tilted algebra is the endomorphism algebra of a cluster tilting object, is the form End(?)(H)(T)op. A large number of research of such algebra show that there are many good properties, see [BMR1, BMR2, ABS1, KR1].In the type of Dynkin, P. Caldero, F. Chapoton and R. Schiffler [CCS1] have associated a category to the cluster algebra of type An, and proved that this category is equivalent to the cluster category of type An. Later, R. Schiffler constructed a geometric model for cluster category of type Dn in [Sch]. As a generalization of cluster categories, m-cluster categories (?)m(H) were intro-duced in [Th]. They are defined as the orbit category Db(H)/τ-1[m]. An object T in (?)m(H) is an m-cluster tilting object if and only if the following hold:Let T be an m-cluster tilting object. B=End(?)m(H)(T)op is the m-cluster-tilted algebra. Recently, there are a lot of reseach about these kinds of algebras, see [BaMl, BaM2, HJ1, HJ2, KR1, KR2, IY, P, Wr, Zh3, ZZ].In this thesis, first, we investigate cluster-tilted algebras of type Dn, classify the quivers of cluster-tilted algebras of type Dn, and moreover, we give a necessary and sufficient condition for the isomorphism between two cluster-tilted algebras of type Dn. Second, we investigate m-cluster-tilted algebras of type An, give a necessary and sufficient condition for the isomorphism between two connected m-cluster-tilted algebras of type An. This thesis is arranged as follows.In Chapter 1, we recall some definitions and basic results needed for our reseach, give some recent developments in this dissertation and make a systemic exposition of our main results.In Chapter 2, by using the geometric model for cluster category of type Dn by R. Schiffler in [Sch], we investigate the cluster-tilted algebras of type Dn. First, we give an explicit description of the equivalence class of triangulations of the category of tagged arces of punctured polygon Pn. As an application, we obtain the mutation classes of quivers of type Dn, and deduce all the quivers of cluster-tilted algebras of type Dn, and moreover, we also give an explicit description for the relation ideal. Next, we give a necessary and sufficient condition for the isomorphism between two cluster-tilted algebras of type Dn. Our main results are as follows.Theorem 2.2.5. The quiver Q is a quiver of cluster-tilted algebras of type Dn if and only if Q belongs to one of type 1, type 2, type 3 or type 4 in section 2.2. Theorem 2.3.1. Let T and T' be two cluster tilting objects in the cluster category (?)(H) of type Dn(n≥5),Γ=End(?)(H)(T)op andΓ'=End(?)(H)(T')op be the corresponding cluster-tilted algebras. ThenΓis isomorphic toΓ' if and only if T=τiT' or T=στjT' for some integers i and j, whereτis the Auslander-Reiten translation,σis the automorphism of (?)(H) defined in section 2.3.In Chapter 3, we use the geometric model for m-cluster category of type An by K. Baur and R. Marsh to investigate the m-cluster-tilted algebras of type An. We prove that the tilting graph of m-cluster category of type An is connected, and obtain a necessary and sufficient condition for the connected m-cluster-tilted algebras of type An, and moreover, give a necessary and sufficient condition for the isomorphism between two connected m-cluster-tilted algebras of type An. Our main results are as follows.Theorem 3.2.7. LetΣm be the tilting graph of an m-cluster category of type An. ThenΣm is connected.Theorem 3.2.9. Let T be an m-cluster tilting object in m-cluster category of type An, B=End(?)m(H)(T)op be the corresponding m-cluster-tilted algebra. Then B is connected if and only if for any indecomposable direct summand M of T, there exists another indecomposable direct summand N of T such that N and M lie on a same ray or coray.Theorem 3.2.10. LetΣm' be the tilting subgraph comsisting of connected m-cluster tilting objects in the m-cluster category of type An. ThenΣm' is connected.Theorem 3.3.3. Let T and T' are two connected m-cluster tilting objects in the cluster category (?)m(H) of type An, B=End(?)m(H)(T)op and B'=End(?)m(H)(T')op be the corresponding m-cluster-tilted algebras. Then B is isomorphic to B' if and only if T=T'[i] for some integer i.