PUB - Publikationen an der Universität Bielefeld

Chapter 2:
We construct two functors from the submodule category of a self-injective representation-finite algebra $\Lambda$ to the module category of the stable Auslander algebra of $\Lambda$. They factor through the module category of the Auslander algebra of $\Lambda$. Moreover they induce equivalences from the quotient categories of the submodule category modulo their respective kernels and said kernels have finitely many indecomposable objects up to isomorphism. We show how this interacts with an idempotent recollement of the module category of the Auslander algebra of $\Lambda$, and get a characterisation of the self-injective Nakayama algebras as a byproduct.

Chapter 3:
We recall how dense orbits of a general linear group acting on quiver flag varieties correspond to rigid objects in monomorphism categories. In order to identify rigid objects via the AR-formula we show how the AR-translate of a representation category of a quiver can be used to calculate the AR-translates of objects in the monomorphism categories of the corresponding path algebra.
We also illustrate other methods to find rigid objects in monomorphism categories; via a long exact sequence, and so called ext-directed decompositions.

Chapter 4:
We introduce the notion of a quiver-graded Richardson orbit, generalising the notion of a dense orbit of a parabolic subgroup of a general linear group acting on the nilpotent radical of its Lie algebra. In this generalised setting dense orbits do not exist in general. We introduce the nilpotent quiver algebra, which is simultaneously left strongly quasi-hereditary and right ultra strongly quasi-hereditary.
We show there is a one-to-one correspondence between rigid objects in the subcategory of standard filtered modules up to isomorphism and quiver-graded Richardson orbits.