On homomorphisms from Ringel-Hall algebras to quantum cluster algebras
arXiv:1402.4213
Abstract
In \cite{rupel3},the authors defined algebra homomorphisms from the dual Ringel-Hall algebra of certain hereditary abelian category to an appropriate -polynomial algebra. In the case that is the representation category of an acyclic quiver, we give an alternative proof by using the cluster multiplication formulas in \cite{DX}. Moreover, if the underlying graph of is bipartite and the matrix associated to the quiver is of full rank, we show that the image of the algebra homomorphisms is in the corresponding quantum cluster algebra.
10 pages