Green Formula in Hall Algebras and Cluster Algebras

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 Green’s formula. In this paper, three forms of Green’s formula are highlighted, (1) the original form of Green’s formula [Gre][Rin2], (2) the degeneration form of Green’s formula [DXX] and (3) the projective form of Green’s formula [XX2] i.e. Green formula with a C-action. The original Green’s formula supplies the comultiplication structure on Ringel-Hall algebras. This compatibility theorem for multiplication and comultiplication on Ringel-Hall algebras deals with the symmetric relation between extensions and flags in the module category of a hereditary algebra. It provides the quantum group a Hopf algebra structure and the Drinfeld double in a global way (see [Gre], [X] and [Ka]).The second and third section contribute to these results. The degenerated Green formula can be viewed as the Green formula over the complex field. We found that it holds for any algebra given by quiver with relations, not only hereditary algebras. We write the formula in a geometric version of the original Green formula, although we know that it essentially agrees with the restriction functor given by Lusztig in [Lu1]. It is applied to provide the geometric realization of the comultiplication of universal enveloping algebras. Section 4 is concerned with these results. The projective version of Green formula has its independent interest. We give its expression and explain its meaning in Section 6. It is applied to prove the cluster multiplication theorem which extends the Caldero-Keller formula [CK1]. Section 7 is used to explain the proof in some details. There is a key ingredient contributing to the above application to cluster categories, i.e. 2-Calabi-Yau property for the cluster categories. In the last section, we extend the multiplication formula in [GLS] for preprojective algebras to any 2-Calabi-Yau algebras. Since our main concern in this article is around various forms of Green formula and their applications, also due to lack of space and knowledge, we do not include many important topics related to cluster algebras and Hall algebras. For cluster algebras we refer to [FZ3] for an excellent survey and we refer to [DX2] and [Sc] for further study on Ringel-Hall algebras.