Maximal Green Sequences for Preprojective Algebras

Maximal green sequences were introduced as combinatorical counterpart for Donaldson-Thomas invariants for 2-acyclic quivers with potential by B. Keller. We take the categorical notion and introduce maximal green sequences for hearts of bounded t-structures of triangulated categories that can be tilted indefinitely. We study the case where the heart is the category of modules over the preprojective algebra of a quiver without loops. The combinatorical counterpart of maximal green sequences for Dynkin quivers are maximal chains in the Hasse quiver of basic support τ-tilting modules. We show that a quiver has a maximal green sequence if and only if it is of Dynkin type. More generally, we study module categories for finite-dimensional algebras with finitely many bricks.     

A New Characterization of Hereditary Algebras

In this short article, we prove that a finite dimensional algebra is hereditary if and only if there is no loop in its ordinary quiver and every τ-tilting module is tilting.     

<Emphasis Type="Italic">G</Emphasis>-stable support <Emphasis Type="Italic">τ</Emphasis>-tilting modules

Motivated by τ-tilting theory developed by T. Adachi, O. Iyama, I. Reiten, for a finite-dimensional algebra Λ with action by a finite group G; we introduce the notion of G-stable support τ-tilting modules. Then we establish bijections among G-stable support τ-tilting modules over Λ; G-stable two-term silting complexes in the homotopy category of bounded complexes of finitely generated projective Λ-modules, and G-stable functorially finite torsion classes in the category of finitely generated left Λ-modules. In the case when Λ is the endomorphism of a G-stable cluster-tilting object T over a Hom-finite 2-Calabi-Yau triangulated category C with a G-action, these are also in bijection with G-stable cluster-tilting objects in C. Moreover, we investigate the relationship between stable support τ-tilitng modules over Λ and the skew group algebra ΛG     

<Emphasis Type="Italic">τ</Emphasis>-Tilting Modules Over One-Point Extensions by a Projective Module

Let A be the one point extension of an algebra B by a projective B-module. We prove that the extension of a given support τ-tilting B-module is a support τ-tilting A-module; and, conversely, the restriction of a given support τ-tilting A-module is a support τ-tilting B-module. Moreover, we prove that there exists a full embedding of quivers between the corresponding poset of support τ-tilting modules.     

Frobenius–Perron theory of modified ADE bound quiver algebras

The Frobenius–Perron dimension for an abelian category was recently introduced in [5]. We apply this theory to the category of representations of the finite-dimensional radical square zero algebras associated to certain modified ADE graphs. In particular, we take an ADE quiver with arrows in a certain orientation and an arbitrary number of loops at each vertex. We show that the Frobenius–Perron dimension of this category is equal to the maximum number of loops at a vertex. Along the way, we introduce a result which can be applied in general to calculate the Frobenius–Perron dimension of a radical square zero bound quiver algebra. We use this result to introduce a family of abelian categories which produce arbitrarily large irrational Frobenius–Perron dimensions.     

Maximal Rigid Objects as Noncrossing Bipartite Graphs

We classify the maximal rigid objects of the Σ² τ-orbit category ${\mathcal{C}}(Q)$ of the bounded derived category for the path algebra associated to a Dynkin quiver Q of type A, where τ denotes the Auslander-Reiten translation and Σ² denotes the square of the shift functor, in terms of bipartite noncrossing graphs (with loops) in a circle. We describe the endomorphism algebras of the maximal rigid objects, and we prove that a certain class of these algebras are iterated tilted algebras of type A.     

HILBERT BASIS QUIVERS

Abstract

A quiver G (= directed multigraph, loops and parallel edges are allowed) is called a Hilbert basis quiver (HBQ) if a certain path algebra R[G] over a ring R is right noetherian provided R does. Such path algebras can be considered as generalized polynomial rings over R. There is the following characterization:

A quiver with a finite number of vertices is HBQ iff its set of edges is finite and its nontrivial path components are elementary cycles, up to parallel edges, which in addition are sink sets (i.e. there is no path leaving the component).

To prove this categorical methods are used.     

Infinite Components of An Auslander-Reiten Quiver with Finite DTr-Orbits

Abstract

Let A be a basic connected finite dimensional algebra over an algebraically closed field. We show that if Γ is an infinite connected component of the Auslander-Reiten quiver Γ_A of A in which each Γ_A-orbit contains only finitely many vertices, then the number of indecomposable direct summands of the middle term of any mesh, whose starting vertex belongs to the infinite stable part of Γ, is less than or equal to 3. Moreover, if the nonstable vertices belong to τ_A-orbits of exceptional projectives in Γ, then Γ can be obtained from a stable tube by a finite number of multiple coray-ray insertions of type α?γ and multiple coray-ray insertions of type α?γ.     

Noncommutative maximal ergodic theorems for positive contractions

Let M be a von Neumann algebra equipped with a normal semifinite faithful trace τ. Let T be a positive linear contraction on M such that τ○T?τ and such that the numerical range of T as an operator on L₂(M) is contained in a Stoltz region with vertex 1. We show that Junge and Xu's noncommutative Stein maximal ergodic inequality holds for the powers of T on Lp(M), 1<p?∞. We apply this result to obtain the noncommutative analogue of a recent result of Cohen concerning the iterates of the product of a finite number of conditional expectations.     

The Graded Preprojective Algebra of a Quiver

The paper introduces a new grading on the preprojective algebraof an arbitrary locally finite quiver. Viewing the algebra asa left module over the path algebra, the author uses the gradingto give an explicit geometric construction of a canonical collectionof exact sequences of its submodules. If a vertex of the quiveris a source, the above submodules behave nicely with respectto the corresponding reflection functor. It follows that whenthe quiver is finite and without oriented cycles, the canonicalexact sequences are the almost split sequences with preprojectiveterms, and the indecomposable direct summands of the submodulesare the non-isomorphic indecomposable preprojective modules.The proof extends that given by Gelfand and Ponomarev in thecase when the finite quiver is a tree. 2000 Mathematics SubjectClassification 16G10, 16G70.     

Local and 2-local derivations on noncommutative Arens algebras

The paper is devoted to so-called local and 2-local derivations on the noncommutative Arens algebra L ^ω(M,τ) associated with a von Neumann algebra M and a faithful normal semi-finite trace τ. We prove that every 2-local derivation on L ^ω(M,τ) is a spatial derivation, and if M is a finite von Neumann algebra, then each local derivation on L ^ω(M,τ) is also a spatial derivation and every 2-local derivation on M is in fact an inner derivation.     

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.     

Invariant bilinear forms on a vertex algebra

In this paper we construct a linear space that parameterizes all invariant bilinear forms on a given vertex algebra with values in a arbitrary vector space. Also we prove that every invariant bilinear form on a vertex algebra is symmetric. This is a generalization of the result of Li (J. Pure Appl. Algebra 96(3) (1994) 279), who proved this for the case when the vertex algebra is non-negatively graded and has finite dimensional homogeneous components.As an application, we introduce a notion of a radical of a vertex algebra. We prove that a radical-free vertex algebra A is non-negatively graded, and its component A₀ of degree 0 is a commutative associative algebra, so that all structural maps and operations on A are A₀-linear. We also show that in this case A is simple if and only if A₀ is a field.     

Generic variables in acyclic cluster algebras

Let Q be an acyclic quiver. We introduce the notion of generic variables for the coefficient-free acyclic cluster algebra A(Q). We prove that the set G(Q) of generic variables contains naturally the set M(Q) of cluster monomials in A(Q) and that these two sets coincide if and only if Q is a Dynkin quiver. We establish multiplicative properties of these generic variables analogous to multiplicative properties of Lusztig’s dual semicanonical basis. This allows to compute explicitly the generic variables when Q is a quiver of affine type. When Q is the Kronecker quiver, the set G(Q) is a Z-basis of A(Q) and this basis is compared to Sherman-Zelevinsky and Caldero-Zelevinsky bases.     

ν-Stable τ-Tilting Modules

Inspired by τ-tilting theory [3 Adachi , T. , Iyama , O. , Reiten , I. ( 2014 ). τ-tilting theory . Compos. Math. 150 ( 3 ): 415 – 452 .[Crossref], [Web of Science ®] , [Google Scholar]], we introduce the notion of ν-stable support τ-tilting modules. For any finite dimensional selfinjective algebra Λ, we give bijections between two-term tilting complexes in K ^b (proj Λ), ν-stable support τ-tilting Λ-modules, and ν-stable functorially finite torsion classes in modΛ. Moreover, these objects correspond bijectively to selfinjective cluster tilting objects in 𝒞 if Λ is a 2-CY tilted algebra associated with a Hom-finite 2-CY triangulated category 𝒞. We also study some properties of support τ-tilting modules over 2-CY tilted algebras, and we give a necessary condition such that algebras are 2-CY tilted in terms of support τ-tilting modules.     

Relative rigid objects in extriangulated categories

In this paper, we study a close relationship between relative cluster tilting theory in extriangulated categories and τ-tilting theory in module categories. Our main results show that relative rigid objects are in bijection with τ-rigid pairs, and also relative maximal rigid objects with support τ-tilting pairs under some assumptions. These results generalize the work by Adachi-Iyama-Reiten, Yang-Zhu and Fu-Geng-Liu. In addition, we introduce mutation of relative maximal rigid objects and show that any basic relative almost maximal rigid object has exactly two non-isomorphic indecomposable complements. All results highlight new phenomena when they applied to exact categories.     

A quiver presentation for Solomon's descent algebra

The descent algebra Σ(W) is a subalgebra of the group algebra QW of a finite Coxeter group W, which supports a homomorphism with nilpotent kernel and commutative image in the character ring of W. Thus Σ(W) is a basic algebra, and as such it has a presentation as a quiver with relations. Here we construct Σ(W) as a quotient of a subalgebra of the path algebra of the Hasse diagram of the Boolean lattice of all subsets of S, the set of simple reflections in W. From this construction we obtain some general information about the quiver of Σ(W) and an algorithm for the construction of a quiver presentation for the descent algebra Σ(W) of any given finite Coxeter group W.