Connectednes of the Cluster Complex

Abstract

The cluster complex of a basic hereditary artin algebra is defined, by constructing a poset T_H. A proof is given that the poset T_H is an abstract simplicial polytope. We define tilting modules, the Bongartz complement, and the relative Bongartz complement and look at the properties. We define approximations and extension and give proofs of the basic properties. We also define perpendicular categories, and give a brief overview of AR-theory.