Abstract not found

Abstract. We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation theory of hereditary algebras. As an application of this, we prove a generalised version of so-called APR-tilting.

Abstract not found

Abstract. We prove that mutation of cluster-tilting objects in triangulated 2-Calabi-Yau categories is closely connected with mutation of quivers with potentials. This gives a close connection between 2-CY-tilted algebras and Jacobian algebras associated with quivers with potentials. We show that cluster-tilted algebras are Jacobian and also that they are determined by their quivers. There are similar results when dealing with tilting modules over 3-CY algebras. The nearly Morita equivalence for 2-CY-tilted algebras is shown to hold for the finite length modules over Jacobian algebras.

Abstract. Let A be an algebra over a field k, and denote by D b (Mod A) the bounded derived category of left A-modules. The derived Picard group DPick(A) is the group of triangle auto-equivalences of D b (Mod A) induced by tilting complexes. In [Ye2] we proved that DPick(A) parameterizes the isomorphism classes of dualizing complexes over A. Also when A is either commutative or local, DPick(A) ∼ = Pick(A) × Z, where Pick(A) is the noncommutative Picard group (the group of Morita equivalences). In this paper we study the group DPick(A) when A = k⃗ ∆ is the path algebra of a finite quiver ⃗ ∆. We obtain general results on the structure of DPick(A), as well as explicit calculations for the Dynkin and affine quivers, and for some wild quivers with multiple arrows. Our method is to construct a representation of DPick(A) on a certain infinite quiver. This representation is faithful when ⃗ ∆ is a tree, and then DPick(A) is discrete. Otherwise a connected linear algebraic group can occur as a factor

this paper k denotes an algebraically closed field and all algebras, categories and functors are assumed to be k-linear; in addition all algebras are assumed to be connected, basic, finite-dimensional and with identity. In [15, Theorem 4.2] Rickard classified Brauer tree algebras up to derived equivalence, in connection with Brou'e's problem [5]. In [1, Theorem 5.8] we extended this classification to representation-finite selfinjective algebras (6= k) of class A n using a covering technique for derived equivalence developed there (Cf. Membrillo-Hernandez [13]). In this paper we will further extend this classification to the class RFSS of representation-finite selfinjective standard algebras (6= k). Moreover we will show that in the class of representation-finite selfinjective algebras, any standard algebra is not stably equivalent to any non-standard algebra. It immediately follows from this that the class RFSS is closed under stable equivalence. Note that derived equivalent selfinjective algebras are always stably equivalent by Keller-Vossieck [12, 2.3 Example] or Rickard [15, Theorem 2.1]. As a corollary of our results we see that the converse of this statement holds if one of the algebras is in the class RFSS. Namely, for any 2 RFSS, a selfinjective algebra is derived equivalent to iff it is stably equivalent to . Section 1 is devoted to a preparation. In particular we gave the precise form of induced display-functors of repetitions for a necessary class of algebras for later use. In section 2 we state our main results, which also give "complete invariants" under derived equivalence for RFSS and a complete set of representatives of the derived equivalence classes in RFSS. In section 3 we introduce computations in a formal additive hull of a spectroid to make the who...

Abstract. In this article we study Cohen-Macaulay modules over one-dimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological method using higher almost split sequences and results from birational geometry. We obtain a large class of 2-CY tilted algebras which are finite dimensional symmetric and satisfies τ 2 = id. In particular, we compute 2-CY tilted algebras for simple/minimally elliptic curve singuralities.

Abstract. We show that the stable module categories of certain selfinjective algebras of finite representation type having tree class An are actually u-cluster categories. Since their introduction in [6], [7], cluster categories have become a central topic in representation theory. They provide the framework for the representation-theoretic approach to the highly successful theory of cluster algebras, as introduced by Fomin and Zelevinsky [11].

Abstract not found

Abstract. We describe recent work on preprojective algebras and moduli spaces of their representations. We give an analogue of Kac’s Theorem, characterizing the dimension types of indecomposable coherent sheaves over weighted projective lines in terms of loop algebras of Kac-Moody Lie algebras, and explain how it is proved using Hall algebras. We discuss applications to the problem of describing the possible conjugacy classes of sums and products of matrices in known conjugacy classes.