We study the cluster algebras arising from cluster tubes with rank bigger than 1. Cluster tubes are 2−Calabi-Yau triangulated categories which contain no cluster tilting objects, but maximal rigid objects. Fix a maximal rigid object T in the cluster tube Γn of rank n (n > 1). For any indecomposable rigid object M in Γn, we define an analogous XM of Caldero-Chapton’s formula (or Palu’s cluster c...

We study a class of infinite dimensional Lie algebras called generalized Witt algebras (in one variable). These include the classical Witt algebra and the centerless Virasoro algebra as important examples. We show that any such generalized Witt algebra is a semisimple, indecomposable Lie algebra which does not contain any abelian Lie subalgebras of dimension greater than one. We develop an inva...

Let A be a hereditary algebra over an algebraically closed field. We prove that an exact fundamental domain for the m-cluster category Cm(A) of A is the m-left part Lm(A (m)) of the m-replicated algebra of A. Moreover, we obtain a one-toone correspondence between the tilting objects in Cm(A) (that is, the m-clusters) and those tilting modules in modA(m) for which all non projective-injective di...

We study triples of coisotropic or isotropic subspaces in symplectic vector spaces; particular, we classify indecomposable structures this kind. The classification depends on the ground field, which assume only to be perfect and not characteristic 2. Our work uses theory representations partially ordered sets with (order reversing) involution; for (co)isotropic triples, relevant poset is “2 + 2...

By counting with triangles and the octohedral axiom, we find a direct way to construct the Ringel-Hall algebra associated to a triangulated category with (left) homological-finite condition. Our formula is a refinement of that in Peng-Xiao [5] and provides a modified version of Toën’s formula in [7]. 1. Calculation with triangles Given a finite field k with q elements, let C be a k-additive tri...

Assuming the generalized continuum hypothesis we construct arbitrarily big indecomposable Banach spaces. i.e., such that whenever they are decomposed as X ⊕ Y , then one of the closed subspaces X or Y must be finite dimensional. It requires alternative techniques compared to those which were initiated by Gowers and Maurey or Argyros with the coauthors. This is because hereditarily indecomposabl...

In [9, 10], Hartshorne gives an account of the known vector bundles of "small" rank on P" and also on the punctured spectrum of a regular local ring. Specifically, he remarks on various constructions of rank two vector bundles on P", for n ^ 4, and Horrocks' examples [12] of rank three bundles on P as well. The result of Evans and Griffith [7] on syzygies shows that the existence of rank two in...