Left-sided quasi-invertible bimodules over Nakayama algebras

Bimodules over triangular Nakayama algebras that give stable equivalences of Morita type are studied here. As a consequence one obtains that every stable equivalence of Morita type between triangular Nakayama algebras is a Morita equivalence.     

Classifying tame blocks and related algebras up to stable equivalences of Morita type

We contribute to the classification of finite dimensional algebras under stable equivalence of Morita type. More precisely we give a classification of Erdmann’s algebras of dihedral, semi-dihedral and quaternion type and obtain as byproduct the validity of the Auslander-Reiten conjecture for stable equivalences of Morita type between two algebras, one of which is of dihedral, semi-dihedral or quaternion type.     

A note on stable equivalences of Morita type

We investigate when an exact functor --Γ which induces a stable equivalence is part of a stable equivalence of Morita type. If Λ and Γ are finite dimensional algebras over a field k whose semisimple quotients are separable, we give a necessary and sufficient condition for this to be the case. This generalizes a result of Rickard’s for self-injective algebras. As a corollary, we see that the two functors given by tensoring with the bimodules in a stable equivalence of Morita type are right and left adjoints of one another, provided that these bimodules are indecomposable. This fact has many interesting consequences for stable equivalences of Morita type. In particular, we show that a stable equivalence of Morita type induces another stable equivalence of Morita type between certain self-injective algebras associated to the original algebras. We further show that when there exists a stable equivalence of Morita type between Λ and Γ, it is possible to replace Λ by a Morita equivalent k-algebra Δ such that Γ is a subring of Δ and the induction and restriction functors induce inverse stable equivalences.     

Stable equivalences of Morita type and stable Hochschild cohomology rings

We prove that a stable equivalence of Morita type between finite dimensional algebras preserves the stable Hochschild cohomology rings, that is, Hochschild cohomology rings modulo the projective center, thus generalizing the results of Pogorzały and Xi.     

Construction of stable equivalences of Morita type for finite-dimensional algebras I

In the representation theory of finite groups, the stable equivalence of Morita type plays an important role. For general finite-dimensional algebras, this notion is still of particular interest. However, except for the class of self-injective algebras, one does not know much on the existence of such equivalences between two finite-dimensional algebras; in fact, even a non-trivial example is not known. In this paper, we provide two methods to produce new stable equivalences of Morita type from given ones. The main results are Corollary 1.2 and Theorem 1.3. Here the algebras considered are not necessarily self-injective. As a consequence of our constructions, we give an example of a stable equivalence of Morita type between two algebras of global dimension , such that one of them is quasi-hereditary and the other is not. This shows that stable equivalences of Morita type do not preserve the quasi-heredity of algebras. As another by-product, we construct a Morita equivalence inside each given stable equivalence of Morita type between algebras and . This leads not only to a general formulation of a result by Linckelmann (1996), but also to a nice correspondence of some torsion pairs in -mod with those in -mod if both and are symmetric algebras. Moreover, under the assumption of symmetric algebras we can get a new stable equivalence of Morita type. Finally, we point out that stable equivalences of Morita type are preserved under separable extensions of ground fields.

     

Constructions of stable equivalences of Morita type for finite dimensional algebras II

Suppose k is a field. Let A and B be two finite dimensional k-algebras such that there is a stable equivalence of Morita type between A and B. In this paper, we prove that (1) if A and B are representation-finite then their Auslander algebras are stably equivalent of Morita type; (2) The n-th Hochschild homology groups of A and B are isomorphic for all n≥1. A new proof is also provided for Hochschild cohomology groups of self-injective algebras under a stable equivalence of Morita type.     

Morita equivalence of C*-correspondences passes to the related operator algebras

We revisit a central result of Muhly and Solel on operator algebras of C*-correspondences. We prove that (possibly non-injective) strongly Morita equivalent C*-correspondences have strongly Morita equivalent relative Cuntz–Pimsner C*-algebras. The same holds for strong Morita equivalence (in the sense of Blecher, Muhly and Paulsen) and strong Δ-equivalence (in the sense of Eleftherakis) for the related tensor algebras. In particular, we obtain stable isomorphism of the operator algebras when the equivalence is given by a σ-TRO. As an application we show that strong Morita equivalence coincides with strong Δ-equivalence for tensor algebras of aperiodic C*-correspondences.     

Morita theory in deformation quantization

Various aspects of Morita theory of deformed algebras and in particular of star product algebras on general Poisson manifolds are discussed. We relate the three flavours ring-theoretic Morita equivalence, *-Morita equivalence, and strong Morita equivalence and exemplify their properties for star product algebras. The complete classification of Morita equivalent star products on general Poisson manifolds is discussed as well as the complete classification of covariantly Morita equivalent star products on a symplectic manifold with respect to some Lie algebra action preserving a connection.     

On a Lift of an Individual Stable Equivalence to a Standard Derived Equivalence for Representation-Finite Self-injective Algebras

We shall show that every stable equivalence (functor) between representation-finite self-injective algebras not of type (D _3m ,s/3,1) with m2, 3s lifts to a standard derived equivalence. This implies that all stable equivalences between these algebras are of Morita type.     

INVARIANTS UNDER STABLE EQUIVALENCES OF MORITA TYPE

The aim of this article is to study some invariants of associative algebras under stable equivalences of Morita type.First of all,we show that,if two finite-dimensional selfinjective k-algebras are sta...     

Relatively stable equivalences of Morita type for blocks

Based on the fact that the relatively stable category of a p-block B is equivalent to the relatively stable category of its Brauer correspondent b as triangulated category, we introduce the notion of relatively stable equivalence of Morita type and show that there is a relatively stable equivalence of Morita type between B and b. Some invariants under stable equivalence of Morita type can be generalized to this relative case. In particular, we put forward the generalized Alperin–Auslander conjecture and prove it in special cases.     

Higman ideal, stable Hochschild homology and Auslander-Reiten conjecture

Let A and B be two finite dimensional algebras over an algebraically closed field, related to each other by a stable equivalence of Morita type. We prove that A and B have the same number of isomorphism classes of simple modules if and only if their 0-degree Hochschild Homology groups HH ₀(A) and HH ₀(B) have the same dimension. The first of these two equivalent conditions is claimed by the Auslander-Reiten conjecture. For symmetric algebras we will show that the Auslander-Reiten conjecture is equivalent to other dimension equalities, involving the centers and the projective centers of A and B. This motivates our detailed study of the projective center, which now appears to contain the main obstruction to proving the Auslander-Reiten conjecture for symmetric algebras. As a by-product, we get several new invariants of stable equivalences of Morita type.     

On tame weakly symmetric algebras having only periodic modules

We classify (up to Morita equivalence) all tame weakly symmetric finite dimensional algebras over an algebraically closed field having simply connected Galois coverings, nonsingular Cartan matrices and the stable Auslander-Reiten quivers consisting only of tubes. In particular, we prove that these algebras have at most four simple modules.Received: 25 February 2002     

Orbit algebras that are invariant under stable equivalences of Morita type

In this note we show that there are a lot of orbit algebras that are invariant under stable equivalences of Morita type between self-injective algebras. There are also indicated some links between Auslander-Reiten periodicity of bimodules and noetherianity of their orbit algebras.     

On Schur algebras and related algebras V: some quasi-hereditary algebras of finite type

In a recent paper, Erdmann determined which of the Schur algebras S(n, r) have finite representation type and described the finite type Schur algebras up to Morita equivalence. The present paper grew out of a desire to see Erdmann's results in the more general context of algebras which are quasi-hereditary in the sense of Cline et al. (1988). Weconsider here the class of quasi-hereditary algebras which have a duality fixing simples. This class includes the “generalized Schur algebras” defined and studied by the first author, and the Schur algebras themselves in particular. In the first part we describe the possible Morita types of the quasi-hereditary algebras of finite representation type over an algebraically closed field with duality fixing simples. This is then applied, in the second part, to give the block theoretic refinement of Erdmann's results.     

Morita equivalence of dual operator algebras

We consider notions of Morita equivalence appropriate to weak* closed algebras of Hilbert space operators. We obtain new variants, appropriate to the dual algebra setting, of the basic theory of strong Morita equivalence, and new nonselfadjoint analogues of aspects of Rieffel’s W^∗-algebraic Morita equivalence.     

On Stable Equivalence of Tensor Products of Algebras

The paper investigates the following problem. Let bimodules N, M yield a stable equivalence of Morita type between self-injective K-algebras A and E. Further, let bimodules S, T yield a stable equivalence of Morita type between self-injective K-algebras B and F. Then we want to know whether the functor M ? _A  ? ? _B S: mod(A ? _K B ^op ) → mod(E ? _K F ^op ) induces a stable equivalence between A ? _K B ^op and E ? _K F ^op . There is given a reduction of this problem to some smaller subcategories for self-injective algebras. Moreover, new invariants of stable equivalences of Morita type are constructed in a general case of arbitrary finite-dimensional algebras over a field.     

Mauldin-Williams graphs, Morita equivalence and isomorphisms

We describe a method for associating some non-self-adjoint algebras to Mauldin-Williams graphs and we study the Morita equivalence and isomorphism of these algebras.

We also investigate the relationship between the Morita equivalence and isomorphism class of the -correspondences associated with Mauldin-Williams graphs and the dynamical properties of the Mauldin-Williams graphs.

     

Foxby equivalence, complete modules, and torsion modules

Taking the idea from classical Foxby equivalence, we develop an equivalence theory for derived categories over differential graded algebras. Both classical Foxby equivalence and the Morita equivalence for complete modules and torsion modules developed by Dwyer and Greenlees arise as special cases.