A Multiplication Formula for Algebras with 2-calabi-yau Properties

We associate to any object in the nilpotent module category of an algebra with the 2-Calabi-Yau property a character (in the sense of [11]) and prove a multiplication formula for the characters. This formula extends a multiplication formula for the evaluation forms (in particular, dual semicanonical basis) associated to modules over a preprojective algebra given by Geiss, Leclerc and Schröer [6] which is analogous to the cluster multiplication theorem of Caldero and Keller in [2]. Introduction Let Q be a quiver without oriented cycle and modkQ be the category of finitedimensional representations of Q over a field k. The cluster category C(Q) is the orbit category of the bounded derived category D(modkQ) by the autoequivalence F := [1] ◦ τ. Under the framework of cluster categories, Caldero and Keller associated the cluster variables XM to objects M of the cluster category C(Q) which Q is a simply laced Dynkin quiver. Then, they realized the acyclic cluster algebras of finite type by defining a cluster multiplication formula [2] as follows: χ(PExt(M,N))XMXN = ∑ Y (χ(PExt(M,N)Y ) + χ(PExt (N,M)Y ))XY where M,N ∈ C and Y runs through the isoclasses of C. Inspired by their work, Geiss, Leclerc and Schröer proved a multiplication formula for the product of two evaluation forms associated to modules over the preprojective algebra [6]. Let Λ be the preprojective algebra associated to the quiver Q.We denote by Λe the module variety of nilpotent Λ-modules with dimension vector e. For any Λ-module x ∈ Λe, they define a evaluation form δx associated to x satisfying there is a finite set R(e) of Λe with the finite partition Λe = ⊔ x∈R(e)〈x〉. They proved χ(PExt Λ(x , x)) δx′⊕x′′ = ∑