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.     

New results related to a conjecture of Moore

Let Γ be a group, Γ′ be a subgroup of Γ of finite index, and R be a ring with identity. Assume that M is an RΓ-module whose restriction to RΓ′ is projective. Moore’s conjecture: Assume that, for all ${x \in (\Gamma-\Gamma^{\prime})}$ , either there is an integer n such that ${1 \neq x^{n} \in \Gamma^{\prime}}$ or x has finite order and is invertible in R. Then M is also projective over RΓ. In this paper, we consider an analogue of this conjecture for injective modules. It turns out that the validity of the conjecture for injective modules implies the validity of it on projective and flat modules. It is also shown that the conjecture for injective modules is true whenever Γ belongs to Kropholler’s hierarchy ${{\bf LH}\mathfrak{F}}$ . In addition, assume that M is an RΓ-module whose restriction to RΓ′ is Gorenstein projective (resp. injective), it is proved that M is Gorenstein projective (resp. injective) over RΓ whenever Γ′ is a subgroup of Γ of finite index.     

On Hamilton cycles in certain planar graphs

Let G be a 2-connected plane graph with outer cycle X_G such that for every minimal vertex cut S of G with |S| ≤ 3, every component of G\S contains a vertex of X_G. A sufficient condition for G to be Hamiltonian is presented. This theorem generalizes both Tutte's theorem that every 4-connected planar graph is Hamiltonian, as well as a recent theorem of Dillencourt about NST-triangulations. A linear algorithm to find a Hamilton cycle can be extracted from the proof. One corollary is that a 4-connected planar graph with the vertices of a triangle deleted is Hamiltonian. © 1996 John Wiley & Sons, Inc.     

Stability,domination and irredundance in a graph

In a graph G, a set X is called a stable set if any two vertices of X are nonadjacent. A set X is called a dominating set if every vertex of V – X is joined to at least one vertex of X. A set X is called an irredundant set if every vertex of X, not isolated in X, has at least one proper neighbor, that is a vertex of V – X joined to it but to no other vertex of X. Let α′ and α, γ, and Γ, ir and IR, denote respectively the minimum and maximum cardinalities of a maximal stable set, a minimal dominating set, and a maximal irredundant set. It is known that ir ? γ ? α′ ? α ? Γ ? IR and that if G does not contain any induced subgraph isomorphic to K_1,3, then γ = α′. Here we prove that if G contains no induced subgraph isomorphic to K_1,3 or to the graph H of figure 1, then ir = γ = α′. We prove also that if G contains no induced subgraph isomorphic to K_1,3, to H, or to the graph h of figure 3, then Γ = IR. Finally, we improve a result of Bollobas and Cockayne about sufficient conditions for γ = ir in terms of forbidden subgraphs.     

Dominating sets in n-cubes

A perfectdominatingset S of a graph Γ is a set of vertices of Γ such that every vertex of Γ is either in S or is adjacent to exactly one vertex of S. We show that a perfect dominating set of the n-cube Q_n induces a subgraph of Q_n whose components are isomorphic to hypercubes. We conjecture that each of these hypercubes has the same dimension. We then prove that if Q_r is a component of the subgraph induced by S, then n ? r ? 1 or 3 (mod 6). A number of examples are given and connections with Steiner Systems and codes are noted.     

Invariant random subgroups of linear groups

Let Γ < GL _n (_F) be a countable non-amenable linear group with a simple, center free Zariski closure. Let Sub(Γ) denote the space of all subgroups of Γ with the compact, metric, Chabauty topology. An invariant random subgroup (IRS) of Γ is a conjugation invariant Borel probability measure on Sub(Γ). An IRS is called non-trivial if it does not have an atom in the trivial group, i.e. if it is non-trivial almost surely. We denote by IRS⁰(Γ) the collection of all non-trivial IRS on Γ.

Theorem 0.1: With the above notation, there exists a free subgroup F < Γ and a non-discrete group topology on Γ such that for every μ ∈ IRS⁰(Γ) the following properties hold:

μ-almost every subgroup of Γ is open

• F ·Δ = Γ for μ-almost every Δ ∈ Sub(Γ).
• F ∩ Δ is infinitely generated, for every open subgroup. In particular, this holds for μ-almost every Δ ∈ Sub(Γ).
• The map

Φ: (Sub(Γ), μ) → (Sub(F),Φ*μ) Δ → Δ ∩ F is an F-invariant isomorphism of probability spaces.A more technical version of this theorem is valid for general countable linear groups. We say that an action of Γ on a probability space, by measure preserving transformations, is almost surely non-free (ASNF) if almost all point stabilizers are non-trivial.Corollary 0.2: Let Γ be as in the Theorem above. Then the product of finitely many ASNF Γ-spaces, with the diagonal Γ action, is ASNF.Corollary 0.3: Let Γ < GL_n(F) be a countable linear group, A Δ Γ the maximal normal amenable subgroup of Γ — its amenable radical. If μ ∈ IRS(Γ) is supported on amenable subgroups of Γ, then in fact it is supported on Sub(A). In particular, if A(Γ) = <e> then Δ = <e>, μ almost surely.     

The socle of the last term in a minimal injective resolution

Let Λ and Γ be left and right Noetherian rings and Λ U a generalized tilting module with Γ = End( Λ U ). For a non-negative integer k, if Λ U is (k - 2)-Gorenstein with the injective dimensions of Λ U and U Γ being k, then the socle of the last term in a minimal injective resolution of Λ U is non-zero.     

Representation-finite trivial extension algebras

Let A be a finite-dimensional basic connected associative algebra over an algebraically closed field, and $\text{T(A)=A ? DA}$ its trivial extension by its minimal injective cogenerator. We prove that T(A) is representation-finite of Cartan class Δ if and only if A is an iterated tilted algebra of Dynkin class Δ. The proof also yields a construction procedure for iterated tilted algebras of Dynkin type.     

The structure of a kind of connected components with oriented cycles and semi-stable vertices

LetF be an algebraically closed field. A be a finite-dimensional algebra overF,Γ _A be the Auslander-Reiten quiver ofA,Γ be a connected component of Γ _A with oriented cycles and semi-stable vertices and each non-stable vertex In Γ be a projective-injective vertex. The structure of Γ is studied. Projcct supported by Chinese Postdoctor Fund.     

Gorenstein Injective Envelopes of Artinian Modules

Let R be a commutative Noetherian ring and A an Artinian R-module. We prove that if A has finite Gorenstein injective dimension, then A possesses a Gorenstein injective envelope which is special and Artinian. This, in particular, yields that over a Gorenstein ring any Artinian module possesses a Gorenstein injective envelope which is special and Artinian.     

On L-R Smash Products of Hopf Algebras

Let H be a Hopf algebra and A an H-bimodule algebra. This article investigates homological dimensions and Gorenstein dimensions of L-R smash products A?H. Several well-known results are generalized. Moreover, we explore the stability of Gorenstein projective (flat) precovers and Gorenstein injective preenvelopes between the category of left A-modules and the category of left A?H-modules.     

On isometries of matrix spaces

Let A _kbe the group of isometries of the space of n-by-n matrices over reals (resp. complexes, quaternions) with respect to the Ky Fan k-norm (see the Introduction for the definitions). Let Γ⁰ be the group of transformations of this space consisting of all products of left and right multiplications by the elements of SO(n)(resp. U(n), Sp(n)). It is shown that, except for three particular casesA_k coincides with the normalizer of Γ in Δ group of isometries of the above matrix space with respect to the standard inner product. We also give an alternative treatment of the case D = R n = 4k = 2 which was studied in detail by Johnson, Laffey, and Li [4].     

Hamiltonian groups are color-graph-hamiltonian

A group Γ is said to be color -graph -hamiltonian if Γ has a minimal generating set Δ such that the Cayley color graph D_Δ(Γ) is hamiltonian. It is shown that every hamiltonian group is color -graph -hamiltonian.     

ENTIRE CHROMATIC NUMBER AND △-MATCHING OF OUTERPLANE GRAPHS

Let G be an outerplane graph with maximum degree A and the entire chromatic number Xvef（G）. This paper proves that if △ ≥6, then △＋ 1≤Xvef（G）≤△＋ 2, and Xvef （G） = △＋ 1 if and only if G has a matching M consisting of some inner edges which covers all its vertices of maximum degree.     

Cores of Imprimitive Symmetric Graphs of Order a Product of Two Distinct Primes

A retract of a graph Γ is an induced subgraph Ψ of Γ such that there exists a homomorphism from Γ to Ψ whose restriction to Ψ is the identity map. A graph is a core if it has no nontrivial retracts. In general, the minimal retracts of a graph are cores and are unique up to isomorphism; they are called the core of the graph. A graph Γ is G‐symmetric if G is a subgroup of the automorphism group of Γ that is transitive on the vertex set and also transitive on the set of ordered pairs of adjacent vertices. If in addition the vertex set of Γ admits a nontrivial partition that is preserved by G, then Γ is an imprimitive G‐symmetric graph. In this paper cores of imprimitive symmetric graphs Γ of order a product of two distinct primes are studied. In many cases the core of Γ is determined completely. In other cases it is proved that either Γ is a core or its core is isomorphic to one of two graphs, and conditions on when each of these possibilities occurs is given.     

Vertex‐transitive graphs that remain connected after failure of a vertex and its neighbors

A d‐regular graph is said to be superconnected if any disconnecting subset with cardinality at most d is formed by the neighbors of some vertex. A superconnected graph that remains connected after the failure of a vertex and its neighbors will be called vosperian. Let Γ be a vertex‐transitive graph of degree d with order at least d+4. We give necessary and sufficient conditions for the vosperianity of Γ. Moreover, assuming that distinct vertices have distinct neighbors, we show that Γ is vosperian if and only if it is superconnected. Let G be a group and let S?G\{1} with S=S^?1. We show that the Cayley graph, Cay(G, S), defined on G by S is vosperian if and only if G\(S∪{1}) is not a progression and for every non‐trivial subgroup H and every a∈G, If moreover S is aperiodic, then Cay(G, S) is vosperian if and only if it is superconnected. © 2011 Wiley Periodicals, Inc. J Graph Theory 67:124‐138, 2011     

Elementary Bialgebra Properties of Group Rings and Enveloping Rings: An Introduction to Hopf Algebras

This is a slight extension of an expository paper I wrote a while ago as a supplement to my joint work with Declan Quinn on Burnside's theorem for Hopf algebras. It was never published, but may still be of interest to students and beginning researchers.

Let K be a field, and let A be an algebra over K. Then the tensor product A ? A = A ? _K A is also a K-algebra, and it is quite possible that there exists an algebra homomorphism Δ: A → A ? A. Such a map Δ is called a comultiplication, and the seemingly innocuous assumption on its existence provides A with a good deal of additional structure. For example, using Δ, one can define a tensor product on the collection of A-modules, and when A and Δ satisfy some rather mild axioms, then A is called a bialgebra. Classical examples of bialgebras include group rings K[G] and Lie algebra enveloping rings U(L). Indeed, most of this paper is devoted to a relatively self-contained study of some elementary bialgebra properties of these examples. Furthermore, Δ determines a convolution product on Hom _K (A, A), and this leads quite naturally to the definition of a Hopf algebra.     

An improvement of fraisse's sufficient condition for hamiltonian graphs

Let G be a k-connected graph of order n. For an independent set c, let d(S) be the number of vertices adjacent to at least one vertex of S and > let i(S) be the number of vertices adjacent to at least |S| vertices of S. We prove that if there exists some s, 1 ≤ s ≤ k, such that Σ_xiEX d(X\{X_i}) > s(n?1) – k[s/2] – i(X)[(s?1)/2] holds for every independetn set X ={x₀, x₁ ?x_s} of s + 1 vertices, then G is hamiltonian. Several known results, including Fraisse's sufficient condition for hamiltonian graphs, are dervied as corollaries.     

A characterization of alternating groups. II

Let G be a group. A subset X of G is called an A-subset if X consists of elements of order 3, X is invariant in G, and every two non-commuting members of X generate a subgroup isomorphic to A₄ or to A₅. Let X be the A-subset of G. Define a non-oriented graph Γ(X) with vertex set X in which two vertices are adjacent iff they generate a subgroup isomorphic to A₄. Theorem 1 states the following. Let X be a non-empty A-subset of G. (1) Suppose that C is a connected component of Γ(X) and H = 〈C〉. If H ∩ X does not contain a pair of elements generating a subgroup isomorphic to A₅ then H contains a normal elementary Abelian 2-subgroup of index 3 and a subgroup of order 3 which coincides with its centralizer in H. In the opposite case, H is isomorphic to the alternating group A(I) for some (possibly infinite) set I, |I| ≥ 5. (2) The subgroup 〈X^G〉 is a direct product of subgroups 〈C _α〉-generated by some connected components C _α of Γ(X). Theorem 2 asserts the following. Let G be a group and X⊆G be a non-empty G-invariant set of elements of order 5 such that every two non-commuting members of X generate a subgroup isomorphic to A₅. Then 〈XG〉 is a direct product of groups each of which either is isomorphic to A₅ or is cyclic of order 5. Supported by RFBR grant No. 05-01-00797; FP “Universities of Russia,” grant No. UR.04.01.028; RF Ministry of Education Developmental Program for Scientific Potential of the Higher School of Learning, project No. 511; Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1. __________ Translated from Algebra i Logika, Vol. 45, No. 2, pp. 203–214, March–April, 2006.     

Applications of the growth characteristics induced by the spectral distance

Let A be a complex unital Banach algebra. Using a connection between the spectral distance and the growth characteristics of a certain entire map into A, we derive a generalization of Gelfand’s famous power boundedness theorem. Elaborating on these ideas, with the help of a Phragm´en-Lindel¨of device for subharmonic functions, it is then shown, as the main result, that two normal elements of a C^?-algebra are equal if and only if they are quasinilpotent equivalent.