Abstract. There is a particular analogy between combinatorial aspects of cluster algebras and Kac-Moody algebras: roughly speaking, cluster algebras are associated with skew-symmetrizable matrices while Kac-Moody algebras correspond to (symmetrizable) generalized Cartan matrices. Both classes of algebras and the associated matrices have the same classification of finite type objects by the well-known Cartan-Killing types. In this paper, we study an extension of this correspondence to the affine type. In particular, we establish the cluster algebras which are determined by the generalized Cartan matrices of affine type.

Abstract. In the structural theory of cluster algebras, a crucial role is played by a family of integer vectors, called c-vectors, which parametrize the coefficients. It has recently been shown that each c-vector with respect to an acyclic initial seed is a real root of the corresponding root system. In this paper, we obtain an interpretation of this result in terms of symmetric matrices. We show that, for skew-symmetric cluster algebras, the c-vectors associated with any seed defines a quasi-cartan companion for the corresponding exchange matrix, i.e. they form a companion basis, and we establish some basic combinatorial properties. In particular, we show that these vectors define an admissible cut of edges in the associated quivers.

Let V be a finite dimensional vector space over the two element field. We compute orbits for the linear action of groups generated by transvections with respect to a certain class of bilinear forms on V . In particular, we compute orbits that are in bijection with connected components of real double Bruhat cells in semisimple groups, extending results of M. Gekhtman, B.

We compute the list of all minimal 2-infinite diagrams, which are cluster algebraic analogues of extended Dynkin graphs.