Mathematics – Representation Theory
Scientific paper
2009-07-07
Mathematics
Representation Theory
111 pages, doctoral thesis University of Paderborn (2009)
Scientific paper
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.
No associations
LandOfFree
The Hall algebra and the composition monoid does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The Hall algebra and the composition monoid, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The Hall algebra and the composition monoid will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-355186