By using the approach in cite{XX2006} to Hall algebras arising in homologically finite triangulated categories, we find an `almost associative multiplication structure for indecomposable objects in a 2-periodic triangulated category. As an applicatio
n, we give a new proof of the theorem of Peng and Xiao in cite{PX2000} which provides a way of realizing symmetrizable Kac-Moody algebras and elliptic Lie algebras via 2-periodic triangulated categories.
We define evaluation forms associated to objects in a module subcategory of Ext-symmetry generated by finitely many simple modules over a path algebra with relations and prove a multiplication formula for the product of two evaluation forms. It is an
alogous to a multiplication formula for the product of two evaluation forms associated to modules over a preprojective algebra given by Geiss, Leclerc and Schroer in cite{GLS2006}.
The canonical bases of cluster algebras of finite types and rank 2 are given explicitly in cite{CK2005} and cite{SZ} respectively. In this paper, we will deduce $mathbb{Z}$-bases for cluster algebras for affine types $widetilde{A}_{n,n},widetilde{D}$
and $widetilde{E}$. Moreover, we give an inductive formula for computing the multiplication between two generalized cluster variables associated to objects in a tube.
In cite{CK2005} and cite{Hubery2005}, the authors proved the cluster multiplication theorems for finite type and affine type. We generalize their results and prove the cluster multiplication theorem for arbitrary type by using the properties of 2--Ca
labi--Yau (Auslander--Reiten formula) and high order associativity.
The objective of the present paper is to give a survey of recent progress on applications of the approaches of Ringel-Hall type algebras to quantum groups and cluster algebras via various forms of Greens formula. In this paper, three forms of Greens
formula are highlighted, (1) the original form of Greens formula cite{Green}cite{RingelGreen}, (2) the degeneration form of Greens formula cite{DXX} and (3) the projective form of Greens formula cite{XX2007a} i.e. Green formula with a $bbc^{*}$-action.
