42 research outputs found
On degenerations between preprojective modules over wild quivers
We study minimal degenerations between preprojective modules over wild
quivers. Asymptotic properties of such degenerations are studied, with respect
to codimension and numbers of indecomposable direct summands. We provide
families of minimal disjoint degenerations of arbitrary codimension for almost
all wild quivers and show that no such examples exist in the remaining cases.Comment: Proofs improved, several typos corrected. To appear in Journal of
Algebr
Cycle-finite module categories
We describe the structure of module categories of finite dimensional algebras
over an algebraically closed field for which the cycles of nonzero
nonisomorphisms between indecomposable finite dimensional modules are finite
(do not belong to the infinite Jacobson radical of the module category).
Moreover, geometric and homological properties of these module categories are
exhibited
On the complement of the dense orbit for a quiver of type \Aa
Let \Aa_t be the directed quiver of type \Aa with vertices. For each
dimension vector there is a dense orbit in the corresponding representation
space. The principal aim of this note is to use just rank conditions to define
the irreducible components in the complement of the dense orbit. Then we
compare this result with already existing ones by Knight and Zelevinsky, and by
Ringel. Moreover, we compare with the fan associated to the quiver \Aa and
derive a new formula for the number of orbits using nilpotent classes. In the
complement of the dense orbit we determine the irreducible components and their
codimension. Finally, we consider several particular examples.Comment: 16 pages, 9 figure
The Hall algebra and the composition monoid
Let Q be a quiver. M. Reineke and A. Hubery investigated the connection
between the composition monoid, as introduced by M. Reineke, and the generic
composition algebra, as introduced by C. M. Ringel, specialised at q=0. In this
thesis we continue their work. We show that if Q is a Dynkin quiver or an
oriented cycle, then the composition algebra at q=0 is isomorphic to the monoid
algebra of the composition monoid. Moreover, if Q is an acyclic, extended
Dynkin quiver, we show that there exists an epimorphism from the composition
algebra at q=0 to the monoid algebra of the composition monoid, and we describe
its non-trivial kernel.
Our main tool is a geometric version of BGP reflection functors on quiver
Grassmannians and quiver flags, that is varieties consisting of filtrations of
a fixed representation by subrepresentations of fixed dimension vectors. These
functors enable us to calculate various structure constants of the composition
algebra.
Moreover, we investigate geometric properties of quiver flags and quiver
Grassmannians, and show that under certain conditions, quiver flags are
irreducible and smooth. If, in addition, we have a counting polynomial, these
properties imply the positivity of the Euler characteristic of the quiver flag.Comment: 111 pages, doctoral thesis University of Paderborn (2009
Representation Theory of Finite Dimensional Algebras
[no abstract available