ar X iv : m at h / 05 11 38 4 v 2 [ m at h . R T ] 1 9 Ju n 20 06 Applications of BGP - reflection functors : isomorphisms of cluster algebras ∗
نویسنده
چکیده
Given a symmetrizable generalized Cartan matrix A, for any index k, one can define an automorphism associated with A, of the field Q(u1, · · · , un) of rational functions of n independent indeterminates u1, · · · , un. It is an isomorphism between two cluster algebras associated to the matrix A (see section 4 for precise meaning). When A is of finite type, these isomorphisms behave nicely, they are compatible with the BGP-reflection functors of cluster categories defined in [Z1, Z2] if we identify the indecomposable objects in the categories with cluster variables of the corresponding cluster algebras, and they are also compatible with the ”truncated simple reflections” defined in [FZ2, FZ3]. Using the construction of preprojective or preinjective modules of hereditary algebras by Dlab-Ringel [DR] and the Coxeter automorphisms (i.e., a product of these isomorphisms), we construct infinitely many cluster variables for cluster algebras of infinite type and all cluster variables for finite types.
منابع مشابه
ar X iv : m at h / 05 11 38 2 v 2 [ m at h . R T ] 1 9 Ju n 20 06 Equivalences between cluster categories ∗
Tilting theory in cluster categories of hereditary algebras has been developed in [BMRRT] and [BMR]. Some of them are already proved for hereditary abelian categories there. In the present paper, all basic results about tilting theory are generalized to cluster categories of hereditary abelian categories. Furthermore, for any tilting object T in a hereditary abelian category H, we verify that t...
متن کاملar X iv : m at h / 04 05 17 6 v 4 [ m at h . R T ] 1 6 Ju n 20 06 QUANTIZED SYMPLECTIC OSCILLATOR ALGEBRAS OF RANK ONE
A quantized symplectic oscillator algebra of rank 1 is a PBW deformation of the smash product of the quantum plane with Uq(sl2). We study its representation theory, and in particular, its category O.
متن کاملar X iv : m at h / 06 06 28 9 v 1 [ m at h . A G ] 1 2 Ju n 20 06 On correspondences of a K 3 surface with itself . IV
Let X be a K3 surface with a polarization H of the degree H 2 = 2rs, r, s ≥ 1, and the isotropic Mukai vector v = (r, H, s) is primitive. The moduli space of sheaves over X with the isotropic Mukai vector (r, H, s) is again a K3 surface, Y. In [11] the second author gave necessary and sufficient conditions in terms of Picard lattice N (X) of X when Y is isomorphic to X (some important particula...
متن کاملIsomorphisms in unital $C^*$-algebras
It is shown that every almost linear bijection $h : Arightarrow B$ of a unital $C^*$-algebra $A$ onto a unital$C^*$-algebra $B$ is a $C^*$-algebra isomorphism when $h(3^n u y) = h(3^n u) h(y)$ for allunitaries $u in A$, all $y in A$, and all $nin mathbb Z$, andthat almost linear continuous bijection $h : A rightarrow B$ of aunital $C^*$-algebra $A$ of real rank zero onto a unital$C^*$-algebra...
متن کاملar X iv : m at h / 06 06 28 9 v 2 [ m at h . A G ] 1 9 Ju n 20 06 On correspondences of a K 3 surface with itself . IV
Let X be a K3 surface with a polarization H of the degree H 2 = 2rs, r, s ≥ 1, and the isotropic Mukai vector v = (r, H, s) is primitive. Moduli space of sheaves over X with the isotropic Mukai vector (r, H, s) is again a K3 surface, Y. In [12] second author gave necessary and sufficient conditions in terms of Picard lattice N (X) of X when Y is isomorphic to X (important particular cases were ...
متن کامل