Frobenius bimodules and flat-dominant dimensions

We establish relations between Frobenius parts and between flat-dominant dimensions of algebras linked by Frobenius bimodules.This is motivated by the Nakayama conjecture and an approach of MartinezVilla to the Auslander-Reiten conjecture on stable equivalences.We show that the Frobenius parts of Frobenius extensions are again Frobenius extensions.Furthermore,let A and B be finite-dimensional algebras over a field k,and let domdim(_AX)stand for the dominant dimension of an A-module X.If_BM_A is a Frobenius bimodule,then domdim(A)domdim(_BM)and domdim(B)domdim(_AHom_B(M,B)).In particular,if B■A is a left-split(or right-split)Frobenius extension,then domdim(A)=domdim(B).These results are applied to calculate flat-dominant dimensions of a number of algebras:shew group algebras,stably equivalent algebras,trivial extensions and Markov extensions.We also prove that the universal(quantised)enveloping algebras of semisimple Lie algebras are QF-3 rings in the sense of Morita.

Bimodules associated to vertex operator superalgebras

Let V be a vertex operator superalgebra and m,n ∈ 21Z+. We construct an An(V ) -Am(V )-bimodule An,m(V ) which characterizes the action of V from the level m subspace to level n subspace of an admissible V -module. We also construct the Verma type admissible V -module from an Am(V )-module by using

Tilting Bimodules from Tilting Pairs

Tilting pair was introduced by Miyashita in 2001 as a generalization of tilting module. In this paper, we construct a tilting left Endh（C）-right Endh（T）-bimodule for a given tilting pairs （C,T） in modh, where A is an Artin algebra.

L-R smash products for multiplier Hopf algebras

The theory of L-R smash product is extended to multiplier Hopf algebras and a sufficient condition for L-R smash product to be regular multiplier Hopf algebras is given. In particular, the result of the paper implies Delvaux＇s main theorem in the case of smash products.

Twisting theory for weak Hopf algebras

The main aim of this paper is to study the twisting theory of weak Hopf algebras and give an equivalence between the （braided） monoidal categories of weak Hopf bimodules over the original and the twisted weak Hopf algebra to generalize the result from Oeckl （2000）.

L-octo-algebras

L-octo-algebra with 8 operations as the Lie algebraic analogue of octo- algebra such that the sum of 8 operations is a Lie algebra is discussed. Any octo- algebra is an L-octo-algebra. The relationships among L-octo-algebras, L-quadri- algebras, L-dendriform algebras, pre-Lie algebras and Lie algebras are given. The close relationships between L-octo-algebras and some interesting structures like Rota- Baxter operators, classical Yang-Baxter equations and some bilinear forms satisfying certain conditions are given also.

Property T^(**) for C~*-algebras

Extending the notion of property T of finite von Neumann algebras to general yon Neu- mann algebras, we define and study in this paper property T＊＊ for （possibly non-unital） C＊-algebras. We obtain several results of property T＊＊ parallel to those of property T for unital C＊-algebras. Moreover, we show that a discrete group F has property T if and only if the group C＊-algebra C＊（F） （or equivalently, the reduced group C＊-algebra C＊（F）） has property T＊＊. We also show that the compact operators K（g2） has property T＊＊ but co does not have property T＊＊.

Submodules of the groupoid C~*-algebra

Given an r discrete, principal and amenable groupoid, the bijective correspondence between the family of the closed C 0(G 0) bimodules of C *(G) and the family of the open subsets of the groupoid G is established. Moreover they are rigidity.

Artinian rings with Morita duality

Two theorems are established which properly extend the class of artinian rings with Morita duality. A question of Anh about QF rings is answered in the negative.

Local 3-cocycles of von Neumann algebras

We show that every local 3-cocycle of a von Neumann algebra R into an arbitrary unital dual R-bimodule S is a 3-cocycle.

△-tame quasi-hereditary algebras

Let (K, M,H) be an upper triangular bimodule problem. Briistle and Hille showed that the opposite algebra A of the endomorphism algebra of a projective generator P of the matrices category of (K., M, H) is quasi-hereditary, and there is an equivalence between the category of△-good modules of A and Mat(K, M). In this note, based on the tame theorem for bimodule problems, we show that if the algebra A associated with an upper triangular bimodule problem is of△-tame representation type, then the category F(△) has the homogeneous property, i.e. almost all modules in F(△) are isomorphic to their Auslander-Reiten translations. Moreover, if (K, M,H)is an upper triangular bipartite bimodule problem, then A is of△-tame representation type if and only if F(△) is homogeneous.

Homological Behavior of Relative Cotorsionfree and Cosyzygy Modules

As the dual of the Auslander transpose and the resulting k-torsionfree module,the cotranspose and k-cotorsionfree module with respect to a semidualizing bimodule have been introduced recently.In this paper we first investigate the relation between relative k-cotorsionfree modules and relative k-cosyzygy modules.Then we study the extension closure of these two classes of modules.