Finite Normalizing Extensions of FSN Rings

Fully semiprimary Noetherian bimodules and their bimodule extensions are examined. In the presence of incomparability of the link graph of prime ideals, certain bimodule extensions preserve the fully semiprimary property. In particular, a finite normalizing extension ring of a fully semiprimary Noetherian ring is also fully semiprimary as a bimodule over the base ring. It is shown that the extension ring is itself a fully semiprimary ring. An application to crossed products over finite groups is given.     

A categorification of finite-dimensional irreducible representations of quantum \mathfraksl2{\mathfrak{sl}_2} and their tensor products

The purpose of this paper is to study categorifications of tensor products of finite-dimensional modules for the quantum group for . The main categorification is obtained using certain Harish-Chandra bimodules for the complex Lie algebra . For the special case of simple modules we naturally deduce a categorification via modules over the cohomology ring of certain flag varieties. Further geometric categorifications and the relation to Steinberg varieties are discussed.We also give a categorical version of the quantised Schur–Weyl duality and an interpretation of the (dual) canonical bases and the (dual) standard bases in terms of projective, tilting, standard and simple Harish-Chandra bimodules.     

Weakly closed Jordan ideals in nest algebras on Banach spaces

We show that weakly closed Jordan ideals in nest algebras on Banach spaces are associative ideals. The decomposability of finite-rank operators in Jordan ideals and the commutants of bimodules are also investigated.     

Gorenstein projective bimodules via monomorphism categories and filtration categories

We generalize the monomorphism category from quiver (with monomial relations) to arbitrary finite dimensional algebras by a homological definition. Given two finite dimension algebras A and B, we use the special monomorphism category $\mathsf{Mon}(B,A\text{-}\mathsf{Gproj})$ to describe some Gorenstein projective bimodules over the tensor product of A and B. If one of the two algebras is Gorenstein, we give a sufficient and necessary condition for $\mathsf{Mon}(B,A\text{-}\mathsf{Gproj})$ being the category of all Gorenstein projective bimodules. In addition, if both A and B are Gorenstein, we can describe the category of all Gorenstein projective bimodules via filtration categories. Similarly, in this case, we get the same result for infinitely generated Gorenstein projective bimodules.     

Bimodules over nest algebras and Deddens' theorem

We generalize Deddens' theorem for nest algebras in the case of w*-closed nest algebras bimodules. For each such bimodule, we introduce a norm closed sub-bimodule of it, which corresponds to the radical of a nest algebra and describe it in a number of ways, generalizing known facts about nest algebras.

     

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.     

Ideal intrinsic extensions with connections to PI-rings

In this paper, we extend various classical results by Armendariz and Steinberg, Fisher, Kaplansky, Martindale, Posner, and Rowen on semiprime PI-rings. We do this by introducing several new generalizations of the class of semiprime PI-rings. For these new classes, some structure theorems are obtained, and connections to arbitrary semiprime rings are made (e.g., a semiprime ring has a largest essentially closed ideal from some of these classes). Numerous examples are provided to illustrate and delimit our results.     

Radical cube zero weakly symmetric algebras and support varieties

One of our main results is a classification of all the weakly symmetric radical cube zero finite dimensional algebras over an algebraically closed field having a theory of support via the Hochschild cohomology ring satisfying Dade’s Lemma. In the process we give a characterization of when a finite dimensional Koszul algebra has such a theory of support in terms of the graded centre of the Koszul dual.     

Cibils–Rosso's Theorem for Quantum Groupoids

Abstract

We generalize the Cibils–Rosso's theorem for categories of Sweedler's Hopf bimodules to the one for categories of weak entwined bimodules. We show that the weak entwined bimodules are modules over a certain algebra. Our best results are attained for categories of weak Hopf bimodules over quantum groupoids (weak Hopf algebras), as special cases of weak Doi–Hopf bimodules.     

Radical cube zero selfinjective algebras of finite complexity

One of our main results is a classification of all the possible quivers of selfinjective radical cube zero finite-dimensional algebras over an algebraically closed field having finite complexity. In the paper (Erdmann and Solberg, 2011) [5] we classified all weakly symmetric algebras with support varieties via Hochschild cohomology satisfying Dade’s Lemma. For a finite-dimensional algebra to have such a theory of support varieties implies that the algebra has finite complexity. Hence this paper is a partial extension of [5].     

Formally smooth bimodules

The notion of a formally smooth bimodule is introduced and its basic properties are analyzed. In particular it is proven that a B-A bimodule M which is a generator left B-module is formally smooth if and only if the M-Hochschild dimension of B is at most one. It is also shown that modules M which are generators in the category σ[M] of M-subgenerated modules provide natural examples of formally smooth bimodules.     

Representation type of Jordan algebras

The problem of classification of Jordan bimodules over (non-semisimple) finite dimensional Jordan algebras with respect to their representation type is considered. The notions of diagram of a Jordan algebra and of Jordan tensor algebra of a bimodule are introduced and a mapping Qui is constructed which associates to the diagram of a Jordan algebra J the quiver of its universal associative enveloping algebra S(J). The main results are concerned with Jordan algebras of semi-matrix type, that is, algebras whose semi-simple component is a direct sum of Jordan matrix algebras. In this case, criterion of finiteness and tameness for one-sided representations are obtained, in terms of diagram and mapping Qui, for Jordan tensor algebras and for algebras with radical square equals to 0.     

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.     

Equivalences of comodule categories for coalgebras over rings

In this article we defined and studied quasi-finite comodules, the cohom functors for coalgebras over rings. Linear functors between categories of comodules are also investigated and it is proved that good enough linear functors are nothing but a cotensor functor. Our main result of this work characterizes equivalences between comodule categories generalizing the Morita-Takeuchi theory to coalgebras over rings. Morita-Takeuchi contexts in our setting is defined and investigated, a correspondence between strict Morita-Takeuchi contexts and equivalences of comodule categories over the involved coalgebras is obtained. Finally, we proved that for coalgebras over QF-rings Takeuchi's representation of the cohom functor is also valid.     

Tensor product of Hopf bimodules over a group

We describe the monoidal structure of the category of Hopf bimodules of a finite group and we derive a surjective ring map from the Grothendieck ring of the category of Hopf bimodules to the center of the integral group ring. We consider analogous results for the multiplicative structure of the Hochschild cohomology.

     

Weakly closed Jordan ideals in nest algebras on Banach spaces

We show that weakly closed Jordan ideals in nest algebras on Banach spaces are associative ideals. The decomposability of finite-rank operators in Jordan ideals and the commutants of bimodules are also investigated. Author’s address: J. Li and F. Lu, Department of Mathematics, Suzhou University, Suzhou 215006, People’s Republic of China This research was supported by NNSFC (No. 10771154) and PNSFJ (NO. BK2007049).     

Ext groups in the category of bimodules over a simple Leibniz algebra

In this article, we generalize Loday and Pirashvili's [11] computation of the Ext-category of Leibniz bimodules for a simple Lie algebra to the case of a simple (non Lie) Leibniz algebra. Most of the arguments generalize easily, while the main new ingredient is the Feldvoss-Wagemann's cohomology vanishing theorem for semi-simple Leibniz algebras.