CHARACTERIZATION OF NAKAYAMA M-CLUSTER TILTED ALGEBRA OF TYPE AN - Faisal
Cluster categories were defined in order to model the cluster algebras introduced by Fomin-Zelevinski. This theory links cluster algebras and the representation theory of quivers. There is a one-to-one correspondence between the cluster tilting objects of a cluster category and the corresponding cluster variables of a cluster algebra. A generalization of cluster categories are them-cluster categories wheremis a natural number. Some m-cluster categories have a nice geometric description. In the case of Dynkin An diagram where n is a natural number, the cluster category can be identified as regular polygons with (m(n + 1) + 2) vertices. The indecomposable objects of an m-cluster category can be interpreted as the so called m-diagonals of the regular polygons and an m-cluster tilting object correspond to a maximal non crossing set of m-diagonals. This maximal set of m-diagonals is an (m + 2)- angulation of the regular polygon. The endomorphism of an m-cluster tilting object is called m-cluster tilted algebra. Its quiver representation can be obtained from the geometric description of the (m + 2)-angulation. In the case of Dynkin An, finding m-cluster tilting objects and hence m-cluster tilted algebras have become a combinatorial problem. An example of nonsemisimple algebras is the Nakayama algebras. This class of algebras is always of finite representation type. Nakayama algebras are easily recognized by their quiver representation. By using its quiver one can classify Nakayama algebras into two types: that is of An type and cyclic type. In this dissertation we investigate on interconnection between m-cluster tilted algebras and Nakayama algebras, focusing only onm-cluster tilted algebras derived from the endomorphism rings of m-cluster tilting objects in m-cluster categories of type A. In particular, we characterize m-cluster tilted algebras of type An which are Nakayama algebras. In the characterization of m-cluster tilted algebras of type An, as a general result we obtained that these algebras are always bounded by either zero relation or paths of length two. The main result in this dissertation is the characterization of m-cluster tilted algebras of type An which are Nakayama algebras cyclic type and acyclic type. For cyclic type we have complete characterization : connected Nakayama m- cluster tilted algebras only occur when m = n - 2 and are always bounded by all paths of length two. For acyclic type the classification is divided into two subclass: case m > n -2 and case m < n-2. In the first subclass, for m > n -2, connected Nakayama m-cluster tilted algebras are always bound by either the zero ideal or any collection paths of length two. For m = n < 2, connected Nakayama m-cluster tilted algebras are always bounded by either the zero ideal or any collection paths of length two which are not equal to collection of all paths of length two. For the last subclass, in the case m < n - 2, these algebras can always be found when bounded by ideals generated by at most m relations of paths of length two.