Standard Koszul standardly stratified algebras

In this paper, we prove that every standard Koszul (not necessarily graded) standardly stratified algebra is also Koszul. This generalizes a similar result of [3 Ágoston, I., Dlab, V., Lukács, E. (2003). Quasi-hereditary extension algebras. Algebras Represent. Theory 6:97–117.[Crossref], [Web of Science ®] , [Google Scholar]] on quasi-hereditary algebras.     

Piecewise-Koszul algebras

It is a small step toward the Koszul-type algebras.The piecewise-Koszul algebras are, in general,a new class of quadratic algebras but not the classical Koszul ones,simultaneously they agree with both the classical Koszul and higher Koszul algebras in special cases.We give a criteria theorem for a graded algebra A to be piecewise-Koszul in terms of its Yoneda-Ext algebra E(A),and show an A_∞-structure on E(A).Relations between Koszul algebras and piecewise-Koszul algebras are discussed.In particular,our results are related to the third question of Green-Marcos.     

Construct non-graded bi-Frobenius algebras via quivers

Using the quiver technique we construct a class of non-graded bi-Frobenius algebras. We also classify a class of graded bi-Frobenius algebras via certain equations of structure coefficients.     

Tilted algebras and configurations of self-injective algebras of Dynkin type

We give an easier way to calculate a bijection from the set of isoclasses of tilted algebras of Dynkin type Δ to the set of configurations on the translation quiver ?Δ.     

Algebraic properties of zigzag algebras

Abstract

We give necessary and sufficient conditions for zigzag algebras and certain generalizations of them to be (relative) cellular, quasi-hereditary or Koszul.     

Quasi-hereditary covers of higher zigzag algebras of type A

In this paper we define and study some quasi-hereditary covers for higher zigzag algebras of type A. We show how these algebras satisfy three different Koszul properties: they are Koszul in the classical sense, standard Koszul and Koszul with respect to the standard module Δ, according to the definition given in [24]. This last property gives rise to a well defined duality and we compute the Δ-Koszul dual as the path algebra of a quiver with relations.     

Local automorphisms of semisimple algebras and group algebras

Let F be a field of characteristic not 2, and let A be a finite-dimensional semisimple F -algebra. All local automorphisms of A are characterized when all the degrees of A are larger than 1. If F is further assumed to be an algebraically closed field of characteristic zero, K a finite group, F K the group algebra of K over F , then all local automorphisms of F K are also characterized.     

KMS States for generalized Gauge actions on Cuntz-Krieger algebras

Given a zero-one matrix A we consider certain one-parameter groups of automorphisms of the Cuntz-Krieger algebra , generalizing the usual gauge group, and depending on a positive continuous function H defined on the Markov space _A. The main result consists of an application of Ruelles Perron-Frobenius Theorem to show that these automorphism groups admit a single KMS state.*Partially supported by CNPq.     

Finitistic dimension of standardly stratified algebras

We prove that the projectively and the injectively defined finitistic dimensions of a standardly stratified algebra are always finite by giving the optimal bound for these numbers in terms of the number of simple modules.     

Presenting degenerate Ringel-Hall algebras of type B

We give a presentation for the degenerate Ringel-Hall algebras of type B by studying the corresponding generic extension monoid algebras.As an application,it is shown that the degenerate Ringel-Hall algebras of type B admit multiplicative bases.     

Verlinde algebras and the intersection form on vanishing cycles

We prove Zuber's conjecture [Z] establishing connections of the fusion rules of the su( N )_k WZW model of conformal field theory and the intersection form on vanishing cycles of the associated fusion potential.     

The cohomology of certain Hopf algebras associated with -groups

We study the cohomology of a locally finite, connected, cocommutative Hopf algebra over . Specifically, we are interested in those algebras for which is generated as an algebra by and . We shall call such algebras semi-Koszul. Given a central extension of Hopf algebras with monogenic and semi-Koszul, we use the Cartan-Eilenberg spectral sequence and algebraic Steenrod operations to determine conditions for to be semi-Koszul. Special attention is given to the case in which is the restricted universal enveloping algebra of the Lie algebra obtained from the mod- lower central series of a -group. We show that the algebras arising in this way from extensions by of an abelian -group are semi-Koszul. Explicit calculations are carried out for algebras arising from rank 2 -groups, and it is shown that these are all semi-Koszul for .

     

Notes on sums of algebras, subdirect products and subdirectly irreducible algebras

The paper shows that certain sums of algebras admit a representation as subdirect products. As a corollary, one obtains a characterization of subdirectly irreducible algebras. Examples of modes show that the characterization gives a good basis for a detailed investigation of the subdirectly irreducible algebras. Received September 10, 1999; accepted in final form January 3, 2001.     

Symmetry in the vanishing of Ext over stably symmetric algebras

A Frobenius algebra over a field k is called symmetric if the Nakayama automorphism is an inner automorphism. A stably symmetric algebra is defined to be a generalization of a symmetric k-algebra. In this paper we will study symmetry in the vanishing of Ext for such algebras R, namely, for all finitely generated R-modules M and N, for all i0 if and only if for all i0. We show that a certain class of noetherian stably symmetric Gorenstein algebras, such as the group algebra of a finite group and the exterior algebra Λ(kⁿ) when n is odd, have this symmetry using Serre duality. We also show that every exterior algebra Λ(kⁿ), whether n is even or odd, has this symmetry for graded modules using Koszul duality.     

Gromov's translation algebras, growth and amenability of operator algebras

Translation algebras of finitely generated *-algebras of bounded linear operators on a separable Hilbert space are introduced. Two equivalent forms of amenability for finitely generated *-algebras in terms of the existence of Følner sequences are introduced. These are related to the existence of traces on the associated translation algebra and, in the context of C^*-algebras, are related to weak-filterability and to the existence of hypertraces.     

Generalized McKay quivers of rank three

For each finite subgroup G of SLn（C）, we introduce the generalized Cartan matrix AG in view of McKay correspondence from the fusion rule of its natural representation. Using group theory, we show that the generalized Cartan matrices have similar favorable properties such as positive semi- definiteness as in the classical case of affine Cartan matrices. The complete McKay quivers for SL3 （C） are explicitly described and classified based on representation theory.     

Finite group actions on AF algebras obtained by folding the interval

For any finite groupG we construct examples of an AF algebraA and an action byG onA such that the fixed point algebra is not AF. The construction ofA is done by successive foldings and cuttings of the interval in a way originally suggested by Blackadar and, in a different context, by Connes in his talk in Oslo in 1978.