Semi-derived Ringel-Hall algebras and Drinfeld double

Canonical Bases of Singularity Ringel-hall Algebras and Hall Polynomials

In this paper, the singularity Ringel-Hall algebras are defined. A new class of perverse sheaves are shown to have purity property. The canonical bases of singularity RingelHall algebras are constructed. As an application, the existence of Hall polynomials in the tame quiver algebras is proved.

On Ringel-hall Algebras of Tame Hereditary Algebras

On Homomorphisms from Ringel-hall Algebras to Quantum Cluster Algebras

In [1],the authors defined algebra homomorphisms from the dual RingelHall algebra of certain hereditary abelian categoryA to an appropriate q-polynomial algebra. In the case that A is the representation category of an acyclic quiver, we give an alternative proof by using the cluster multiplication formulas in [9]. Moreover, if the underlying graph of Q is bipartite and the matrix B associated t...

Double Hall Algebras and Derived Equivalences

The resulting algebra is known as the Hall algebra HA of A. Hall algebras first appeared in the work of Steinitz [S] and Hall [H] in the case where A is the category of Abelian p-groups. They reemerged in the work of Ringel [R1]-[R3], who showed in [R1] that when A is the category of quiver representations of an A-D-E quiver ~ Q over a finite field Fq, the Hall algebra of A provides a realizati...

Feynman Graphs, Rooted Trees, and Ringel-hall Algebras

We construct symmetric monoidal categoriesLRF ,LFG of rooted forests and Feynman graphs. These categories closely resemble finitary abelian categories, and in particular, the notion of Ringel-Hall algebra applies. The Ringel-Hall Hopf algebras of LRF ,LFG, HLRF ,HLFG are dual to the corresponding Connes-Kreimer Hopf algebras on rooted trees and Feynman diagrams. We thus obtain an interpretation...