Hall代数与合成么半群
The Hall algebra and the composition monoid
摘要：
设Q为a。M. Reineke和A.胡伯里研究了M. Reineke和C. M.林格尔，在q=0时特化。在本论文中，我们继续他们的工作。证明了如果Q是Dynkin环或定向圈，则q=0处的复合代数同构于复合幺半群的幺半群代数.此外，如果Q是一个非循环的扩展Dynkin代数，我们证明了存在从q=0的复合代数到复合幺半群的幺半群代数的满射，并描述了它的非平凡核.我们的主要工具是一个几何版本的BGP反射函子上的固定Grassmannians和标志，即品种组成的固定表示的固定维数向量的子表示的过滤。这些函子使我们能够计算各种结构常数的复合代数。此外，我们还研究了双旗和双Grassmannian的几何性质，证明了在一定条件下，双旗是不可约的和光滑的。如果，此外，我们有一个计数多项式，这些性质意味着积极的欧拉示性数的旗帜。
英文摘要：
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.
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