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.     

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.     

Notes on piecewise-Koszul algebras

The relationships between piecewise-Koszul algebras and other “Koszul-type” algebras are discussed. The Yoneda-Ext algebra and the dual algebra of a piecewise-Koszul algebra are studied, and a sufficient condition for the dual algebra A ^! to be piecewise-Koszul is given. Finally, by studying the trivial extension algebras of the path algebras of Dynkin quivers in bipartite orientation, we give explicit constructions for piecewise-Koszul algebras with arbitrary “period” and piecewise-Koszul algebras with arbitrary “jump-degree”.     

Lie algebras determined by finite valued Auslander-Reiten quivers

Let г denote a connected valued Auslander-Reiten quiver, let ℒ(γ) denote the free abelian group generated by the vertex setγ ₀ and let ℒ(Γ) be the universal cover ofг with fundamental groupG. It is proved that whenγ is a finite connected valued Auslander-Reiten quiver,ℒ(γ) is a Lie subalgebra ofℋ(г), and is just the “orbit” Lie algebra ℒ( )/G, where ℋ (г)₁ is the degenerate Hall algebra ofг and ℒ( )/G is the “orbit” Lie algebra induced by .     

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.     

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 .

     

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.     

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.     

Cluster tilting for higher Auslander algebras

The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representation-finite algebras and Auslander algebras. The n-Auslander–Reiten translation functor τn plays an important role in the study of n-cluster tilting subcategories. We study the category Mn of preinjective-like modules obtained by applying τn to injective modules repeatedly. We call a finite-dimensional algebra Λ n-complete if for an n-cluster tilting object M. Our main result asserts that the endomorphism algebra EndΛ(M) is (n+1)-complete. This gives an inductive construction of n-complete algebras. For example, any representation-finite hereditary algebra Λ(1) is 1-complete. Hence the Auslander algebra Λ(2) of Λ(1) is 2-complete. Moreover, for any n?1, we have an n-complete algebra Λ(n) which has an n-cluster tilting object M(n) such that Λ(n+1)=EndΛ(n)(M(n)). We give the presentation of Λ(n) by a quiver with relations. We apply our results to construct n-cluster tilting subcategories of derived categories of n-complete algebras.     

有限复杂度的Koszul代数

首先给出了Koszul代数的张量积的复杂度,然后研究了Koszul遗传代数上的Koszul单列模,并证明了Koszul遗传代数上的Koszul模M的Koszul合成列在同构意义下是唯一的.     

Nichols algebras of group type with many quadratic relations

We classify Nichols algebras of irreducible Yetter–Drinfeld modules over nonabelian groups satisfying an inequality for the dimension of the homogeneous subspace of degree two. All such Nichols algebras are finite-dimensional, and all known finite-dimensional Nichols algebras of nonabelian group type appear in the result of our classification. We find a new finite-dimensional Nichols algebra over fields of characteristic two.     

Finite GK-dimensional pre-Nichols algebras of super and standard type

We prove that finite GK-dimensional pre-Nichols algebras of super and standard type are quotients of the corresponding distinguished pre-Nichols algebras, except when the braiding matrix is of type super A and the dimension of the braided vector space is three. For these two exceptions we explicitly construct substitutes as braided central extensions of the corresponding pre-Nichols algebras by a polynomial ring in one variable. Via bosonization this gives new examples of finite GK-dimensional Hopf algebras.     

Quotients of Koszul Algebras and 2-d-Determined Algebras

Vatne [13 Vatne , J. E. ( 2012 ). Quotients of Koszul algebras with almost linear resolution. Preprint, arXiv:1103.3572 . [Google Scholar]] and Green and Marcos [9 Green , E. L. , Marcos , E. N. (2011). d-Koszul algebras, 2-d-determined algebras and 2-d-Koszul algebras. J. Pure Appl. Algebra 215(4):439–449.[Crossref], [Web of Science ®] , [Google Scholar]] have independently studied the Koszul-like homological properties of graded algebras that have defining relations in degree 2 and exactly one other degree. We contrast these two approaches, answer two questions posed by Green and Marcos, and find conditions that imply the corresponding Yoneda algebras are generated in the lowest possible degrees.     

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.     

Characterization of Lie higher derivations on triangular algebras

Let $\mathcal{A}$ and $\mathcal{B}$ be unital rings, and $\mathcal{M}$ be an $\left( {\mathcal{A},\mathcal{B}} \right)$ -bimodule, which is faithful as a left $\mathcal{A}$ -module and also as a right $\mathcal{B}$ -module. Let $\mathcal{U} = Tri\left( {\mathcal{A},\mathcal{M},\mathcal{B}} \right)$ be the triangular algebra. In this paper, we give some different characterizations of Lie higher derivations on $\mathcal{U}$ .