Induced and coinduced modules over cluster-tilted algebras

We propose a new approach to study the relation between the module categories of a tilted algebra C and the corresponding cluster-tilted algebra B=C⋉E. This new approach consists of using the induction functor −⊗_CB as well as the coinduction functor D(B⊗_CD−). We show that DE is a partial tilting and a τ-rigid C-module and that the induced module DE⊗_CB is a partial tilting and a τ-rigid B-module. Furthermore, if C=End_AT for a tilting module T over a hereditary algebra A, we compare the induction and coinduction functors to the Buan-Marsh-Reiten functor Hom_CA (T,−) from the cluster-category of A to the module category of B. We also study the question as to which B-modules are actually induced or coinduced from a module over a tilted algebra.