The representation dimension of domestic weakly symmetric algebras

Auslander’s representation dimension measures how far a finite dimensional algebra is away from being of finite representation type. In [1], M. Auslander proved that a finite dimensional algebra A is of finite representation type if and only if the representation dimension of A is at most 2. Recently, R. Rouquier proved that there are finite dimensional algebras of an arbitrarily large finite representation dimension. One of the exciting open problems is to show that all finite dimensional algebras of tame representation type have representation dimension at most 3. We prove that this is true for all domestic weakly symmetric algebras over algebraically closed fields having simply connected Galois coverings.     

π-Complementation in the Unitisation and Multiplication Algebras of a Semiprime Algebra

π-complemented algebras are defined as those (not necessarily associative or unital) algebras such that each annihilator ideal is complemented by other annihilator ideal. For a given semiprime algebra A, we discuss the π-complementation of the unitisation algebra A ¹ of A. Moreover, if in addition the multiplication algebra ?(A) of A is also semiprime, we study the π-complementation in the algebras ?(A) and ?^?(A) (the multiplication ideal of A). In associative setting, we prove that A is π-complemented if and only if ?^?(A) is π-complemented, and that A ¹ π-complemented if and only if ?(A) is π-complemented.     

Derived-tame Tree Algebras

In this note we classify the derived-tame tree algebras up to derived equivalence. A tree algebra is a basic algebra A = kQ/I whose quiver Q is a tree. The algebra A is said to be derived-tame when the repetitive category  of A is tame. We show that the tree algebra A is derived-tame precisely when its Euler form _A is non-negative. Moreover, in this case, the derived equivalence class of A is determined by the following discrete invariants: The number of vertices, the corank and the Dynkin type of _A . Representatives of these derived equivalence classes of algebras are given by the following algebras: the hereditary algebras of finite or tame type, the tubular algebras and a certain class of poset algebras, the so-called semichain-algebras which we introduce below.     

Endoprimal algebras

An algebra A is endoprimal if, for all the only maps from A ^k to A which preserve the endomorphisms of A are its term functions. One method for finding finite endoprimal algebras is via the theory of natural dualities since an endodualisable algebra is necessarily endoprimal. General results on endoprimality and endodualisability are proved and then applied to the varieties of sets, vector spaces, distributive lattices, Boolean algebras, Stone algebras, Heyting algebras, semilattices and abelian groups. In many classes the finite endoprimal algebras turn out to be endodualisable. We show that this fails in general by proving that , regarded as either a bounded semilattice or upper-bounded semilattice is dualisable, endoprimal but not endodualisable. Received May 16, 1997; accepted in final form November 6, 1997.     

The Simple Connectedness of a Tame Weakly Shod Algebra

Abstract

We prove that a tame weakly shod algebra A which is not quasi-tilted is simply connected if and only if the orbit graph of its pip-bounded component is a tree, or if and only if its first Hochschild cohomology group H¹(A) with coefficients in _A A _A vanishes. We also show that it is strongly simply connected if and only if the orbit graph of each of its directed components is a tree, or if and only if H¹(A) = 0 and it contains no full convex subcategory which is hereditary of type 𝔸?, or if and only if it is separated and contains no full convex subcategory which is hereditary of type 𝔸?.     

Compact DG modules and Gorenstein DG algebras

When the base connected cochain DG algebra is cohomologically bounded, it is proved that the difference between the amplitude of a compact DG module and that of the DG algebra is just the projective dimension of that module. This yields the unboundedness of the cohomology of non-trivial regular DG algebras. When A is a regular DG algebra such that H(A) is a Koszul graded algebra, H(A) is proved to have the finite global dimension. And we give an example to illustrate that the global dimension of H(A) may be infinite, if the condition that H(A) is Koszul is weakened to the condition that A is a Koszul DG algebra. For a general regular DG algebra A, we give some equivalent conditions for the Gorensteiness. For a finite connected DG algebra A, we prove that D^c(A) and D^c(A ^op) admit Auslander-Reiten triangles if and only if A and A ^op are Gorenstein DG algebras. When A is a non-trivial regular DG algebra such that H(A) is locally finite, D^c(A) does not admit Auslander-Reiten triangles. We turn to study the existence of Auslander-Reiten triangles in D_lf^b(A) and D_lf^b (A ^op) instead, when A is a regular DG algebra. This work was supported by the National Natural Science Foundation of China (Grant No. 10731070) and the Doctorate Foundation of Ministry of Education of China (Grant No. 20060246003)     

Actions of hopf algebras on fully bounded noetherian rings

Let H be a finite dimensional Hopf algebra acting on a right noetherian algebra A and assume that the trace function [tcirc] : A → A ^H is surjective. Then A is right PBN if and only if so is A ^H . This extends the result of García-del Río for group actions which answered a question of Fisher-Osterburg, and the result of Nǎstǎsescu-Dǎscǎlescu for group graded algebras.     

Finiteness of the strong global dimension of radical square zero algebras

The strong global dimension of a finite dimensional algebra A is the maximum of the width of indecomposable bounded differential complexes of finite dimensional projective A-modules. We prove that the strong global dimension of a finite dimensional radical square zero algebra A over an algebraically closed field is finite if and only if A is piecewise hereditary. Moreover, we discuss results concerning the finiteness of the strong global dimension of algebras and the related problem on the density of the push-down functors associated to the canonical Galois coverings of the trivial extensions of algebras by their repetitive algebras.     

Radical of filters in BL‐algebras

In this paper, the notion of the radical of a filter in BL‐algebras is defined and several characterizations of the radical of a filter are given. Also we prove that A/F is an MV‐algebra if and only if D_s(A) ? F. After that we define the notion of semi maximal filter in BL‐algebras and we state and prove some theorems which determine the relationship between this notion and the other types of filters of a BL‐algebra. Moreover, we prove that A/F is a semi simple BL‐algebra if and only if F is a semi maximal filter of A. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Derived dynkin extensions

The finite dimensional tame hereditary algebras are associated with the extended Dynkin diagrams. An indecomposable module over such an algebra is either preprojective or preinjective or lies in a family of tubes whose tubular type is the corresponding Dynkin diagram. The study of one-point extensions by simple regular modules in such tubes was initiated in [Ri].

We generalise this approach by starting out with algebras which are derived equivalent to a tame hereditary algebra and considering one-point extensions by modules which are simple regular in tubes in the derived category. If the obtained tubular type is again a Dynkin diagram these algebras are called derived Dynkin extensions.

Our main theorem says that a representation infinite algebra is derived equivalent to a tame hereditary algebra iff it is an iterated derived Dynkin extension of a tame concealed algebra. As application we get a new proof of a theorem in [AS] about domestic tubular branch enlargements which uses the derived category instead of combinatorial arguments.     

Geometry of Periodic Modules over Tame Concealed and Tubular Algebras

Let A be a tame concealed or tubular algebra and d the dimension-vector of a periodic module with respect to the action of the Auslander–Reiten translation. We prove that the affine variety mod _A (d) of all A-modules of dimension-vector d is a normal complete intersection. Moreover, we show that a module M in mod _A (d) is nonsingular if and only if Ext _A ²(M,M)=0.     

Algebraic properties of isometries between groups of invertible elements in Banach algebras

We prove that an isometry T between open subgroups of the invertible groups of unital Banach algebras A and B is extended to a real-linear isometry up to translation between these Banach algebras. While a unital isometry between unital semisimple commutative Banach algebras need not be multiplicative, we prove in this paper that if A is commutative and A or B are semisimple, then (T(eA))⁻¹T is extended to an isometric real algebra isomorphism from A onto B. In particular, A−1 is isometric as a metric space to B−1 if and only if they are isometrically isomorphic to each other as metrizable groups if and only if A is isometrically isomorphic to B as a real Banach algebra; it is compared by the example of ?elazko concerning on non-isomorphic Banach algebras with the homeomorphically isomorphic invertible groups. Isometries between open subgroups of the invertible groups of unital closed standard operator algebras on Banach spaces are investigated and their general forms are given.     

Vectorspace categories immersed in directed components

It is known that, given a tame algebra Λ, the Tits form qΛ is weakly non negative

Moreover, the converse has been shown for some families of algebras, but it is not true in general. The purpose of this work is to show that for certain wild vectorspace categories K = Hom(M,B - mod) where B is tame and M is an indecomposable B-module, we have qB[M] strongly indefinite. This gives partial converses of the above theorem.     

Representation theory of the system quiver

Let A be a three-point algebra with Gabriel quiver the system quiver Q, Then, up to isomorphism and duality, A is tame if and only if A is or degenerates to a factor of a tame algebra in the tame Table T if and only if A/rad5 A is tame, and A is wild if and only if A has a wild algebra in the wild Table W as a factor if and only if A is controlled wild.     

On Finite Quantum Groups at −1

This is a contribution to the classification of finite-dimensional pointed Hopf algebras. We are concerned with the case when the group of group-like elements is Abelian of exponent 2. We attach to such a pointed Hopf algebra a generalized simply-laced Cartan matrix; we conjecture that the Hopf algebra is finite-dimensional if and only if the Cartan matrix is of finite type. We prove the conjecture for the types A_n and A_n⁽¹⁾. We obtain the classification of all possible Hopf algebras with Cartan matrix A_n. We use the lifting method developed by Hans-Jürgen Schneider and the first-named author. Presented by S. MontgomeryMathematics Subject Classifications (2000) Primary: 17B37; secondary: 16W30.This work was partially supported by CONICET, Agencia Córdoba Ciencia – CONICOR, FOMEC and Secyt (UNC).     

Character Connes amenability of dual Banach algebras

We study the notion of character Connes amenability of dual Banach algebras and show that if A is an Arens regular Banach algebra, then A^** is character Connes amenable if and only if A is character amenable, which will resolve positively Runde’s problem for this concept of amenability. We then characterize character Connes amenability of various dual Banach algebras related to locally compact groups. We also investigate character Connes amenability of Lau product and module extension of Banach algebras. These help us to give examples of dual Banach algebras which are not Connes amenable.     

Basic Hopf Algebras of Tame Type

In continuation of the articles (Liu J Algebra 299:841–853, 2006; Huang, J Algebra 321:2650–2669, 2009) we classify all finite-dimensional basic Hopf algebras of tame type over an algebraically closed field of characteristic 0 in this paper. As consequences, we show the following statements: (1) the representation dimension of a tame basic Hopf algebra is exactly 3, (2) for a basic Hopf algebra H, if $\textrm{C}(H)\geq 3$ then it is wild. These conclusions verify a folklore conjecture and one of Rickard’s statements for the class of finite-dimensional basic Hopf algebras.     

The Daugavet Property of C*-Algebras and Non-commutative Lp-Spaces

We prove that a C ^*-algebra A or a predual N _* of a von Neumann algebra N has the Daugavet property if and only if A (or N) is non-atomic. We also prove a similar (although somewhat weaker) result for non-commutative L _p-spaces corresponding to non-atomic von Neumann algebras.     

Existence of Gradings on Associative Algebras

In this article, we study the existence of gradings on finite dimensional associative algebras. We prove that a connected algebra A does not have a nontrivial grading if and only if A is basic, its quiver has one vertex, and its group of outer automorphisms is unipotent. We apply this result to prove that up to graded Morita equivalence there do not exist nontrivial gradings on the blocks of group algebras with quaternion defect groups and one isomorphism class of simple modules.