Integral Bases Of Affine Cluster Algebras

Cluster algebras were introduced by S. Fomin and A. Zelevinsky in order to study the problems of total positivity in algebraic groups and bases in quantum groups. The link between acyclic cluster algebras and representation theory of quivers were first revealed by R. Marsh, M. Reineke, and A. Zelevinsky. Then A. Buan, R. Marsh, M. Reineke, I. Reiten and G. Todorov introduced the cluster categories as the categorification of acyclic cluster algebras.Now let Q be a tame quiver then the underlying graph of Q is of affine type Ap,q(p,q∈N),Dn(n≥4,n∈N) or Em(m=6,7,8) and Q contains no oriented cycles. Using the Caldero-Chapton map, we consider the cluster algebras of affine type over the complex field C. We are interested in constructing the integral bases of them. By using the cluster multiplication formulas and the representation theory of tame quivers, the main results are obtained step by step as follows.Firstly, we investigate a special tame quiver D4 whose AR-quiver can be clearly described according to the representation theory of quivers. Because the structure of the AR-quiver of D4 is explicit, we deduce a Z-basis for the cluster algebra of it by directly calculating the product of any two generalized cluster variables according to the cluster multiplication formulas. It is a generalization of the results given by P. Caldero and B. Keller and P. Sherman and A. Zelevinsky. Moreover, we prove that the coefficients of Laurent expansions in generalized cluster variables for D4 are always positive integers. Note that our method is totally different from the one used by P. Caldero and M. Reineke.Secondly, using the representation theory of tame quivers, we construct the Z-bases for cluster algebras of affine type. Note that G. Dupont gives the Z-bases of the cluster algebras for affine type Ap,q(p,q∈N). Comparing the Z-bases we construct with the generic variables given by G. Dupont, we prove the conjecture of "differ-ence property". Moreover, we give inductive formulas for calculating the product of any two generalized cluster variables associated to objects in a tube. From these induc-tive formulas, we obtain the result of the product and give the algebraic combinatorial calculation of Euler characteristic.