Split Sequences Research Articles

A construction of derived equivalent pairs of symmetric algebras

Recently, Hu and Xi have exhibited derived equivalent endomorphism rings arising from (relative) almost split sequences as well as AR-triangles in triangulated categories. We present a broader class of triangles (in algebraic triangulated categories) for which the endomorphism rings of different terms are derived equivalent. We then study applications involving 0-Calabi-Yau triangulated categories. In particular, applying our results in the category of perfect complexes over a symmetric algebra gives a nice way of producing pairs of derived equivalent symmetric algebras. Included in the examples we work out are some of the algebras of dihedral type with two or three simple modules. We also apply our results to stable categories of Cohen-Macaulay modules over odd-dimensional Gorenstein hypersurfaces having an isolated singularity.

Recognition of split-graphic sequences

Abstract Using different definitions of split graphs we propose quick algorithms for the recognition and extremal reconstruction of split sequences among integer, regular, and graphic sequences.

Almost split sequences in the category of complexes of modules

Let Λ be an artin algebra. By letting the Nakayama functor act degree-wise, we define a translation τ in the category of complexes of finitely generated Λ-modules, C(mod Λ). Then we investigate the existence of almost split sequences in the category C(mod Λ). As an application of our results, we see that the full subcategory of D(mod Λ) consisting of complexes isomorphic to perfect complexes admits almost split sequences.

Methods of infinite dimensional Morse theory for geodesics on Finsler manifolds

We prove the shifting theorems of the critical groups of critical points and critical orbits for the energy functionals of Finsler metrics on Hilbert manifolds of H1-curves, and two splitting lemmas for the functionals on Banach manifolds of C1-curves. Two results on critical groups of iterated closed geodesics are also proved; their corresponding versions on Riemannian manifolds are based on the usual splitting lemma by Gromoll and Meyer (1969)). Our approach consists in deforming the square of the Finsler metric in a Lagrangian which is smooth also on the zero section and then in using the splitting lemma for nonsmooth functionals that the author recently developed in Lu (2011, 0000, 2013). The argument does not involve finite-dimensional approximations and any Palais’ result in Palais (1966). As an application, we extend to Finsler manifolds a result by V. Bangert and W. Klingenberg (1983) about the existence of infinitely many, geometrically distinct, closed geodesics on a compact Riemannian manifold.

On selfinjective algebras of finite representation type with maximal almost split sequences

We investigate the structure of finite dimensional selfinjective algebras of finite representation type over an arbitrary field having almost split sequences of modules whose middle term admits the maximal possible number of indecomposable direct summands.

Green Rings of Pointed Rank One Hopf algebras of Nilpotent Type

Let H be a finite dimensional pointed rank one Hopf algebra of nilpotent type. We first determine all finite dimensional indecomposable H-modules up to isomorphism, and then establish the Clebsch-Gordan formulas for the decompositions of the tensor products of indecomposable H-modules by virtue of almost split sequences. The Green ring r(H) of H will be presented in terms of generators and relations. It turns out that the Green ring r(H) is commutative and is generated by one variable over the Grothendieck ring G 0(H) of H modulo one relation. Moreover, r(H) is Frobenius and symmetric with dual bases associated to almost split sequences, and its Jacobson radical is a principal ideal. Finally, we show that the stable Green ring, the Green ring of the stable module category, is isomorphic to the quotient ring of r(H) modulo all projective modules. It turns out that the complexified stable Green algebra is a group-like algebra and hence a bi-Frobenius algebra.

The traces associated with a sharp tridiagonal system

Let K denote a field and let V denote a vector space over K with finite positive dimension. Let A,A⁎ denote a tridiagonal pair on V. Let {θi}i=0d (resp. {θi⁎}i=0d) denote a standard ordering of the eigenvalues of A (resp. A⁎) and for 0≤i≤d let Vi (resp. Vi⁎) be the eigenspace of A (resp. A⁎) associated with θi (resp. θi⁎). It is known that Vi,Vi⁎ have the same dimension. The tridiagonal pair A,A⁎ is said to be sharp whenever dim(V0)=1. For 0≤i≤d, let Ei (resp. Ei⁎) denote the primitive idempotent of A (resp. A⁎) associated with θi (resp. θi⁎). Then Φ=(A;E0,E1,⋯,Ed;A⁎;E0⁎,E1⁎,⋯,Ed⁎) is a tridiagonal system on V. We say Φ is sharp whenever the tridiagonal pair A,A⁎ is sharp. Assume Φ is sharp and let {ζi}i=0d denote the split sequence of Φ. The sequence ({θi}i=0d;{θi⁎}i=0d;{ζi}i=0d) is called the parameter array of Φ. Recently in [3] K. Nomura and P. Terwilliger proposed the following problem: Let Φ denote a sharp tridiagonal system. For 0≤i≤d find each oftr(EiE0⁎),tr(EiEd⁎),tr(Ei⁎E0),tr(Ei⁎Ed) in terms of the parameter array of Φ. In the present paper we solve this problem.

Almost split sequences of the quantum double of a finite group

Let k be a field of characteristic p > 0, and G be a finite group of order divisible by p. We prove that the almost split sequences of the quantum double D(kG) can be constructed from those of group algebras, where the groups run over all centralizer subgroups of representatives of conjugate classes of G. As a special case, we give an application to the quantum double of dihedral groups.

Matroid base polytope decomposition II: Sequences of hyperplane splits

This is a continuation of an early paper Chatelain et al. (2011) [3] about matroid base polytope decomposition. We will present sufficient conditions on a matroid M so its base polytope P(M) has a sequence of hyperplane splits. These yield to decompositions of P(M) with two or more pieces for infinitely many matroids M. We also present necessary conditions on the Euclidean representation of rank three matroids M for the existence of decompositions of P(M) into 2 or 3 pieces. Finally, we prove that P(M1⊕M2) has a sequence of hyperplane splits if either P(M1) or P(M2) also has a sequence of hyperplane splits.

Strong Immersions and Maximum Degree

A graph $H$ is strongly immersed in $G$ if $G$ is obtained from $H$ by a sequence of vertex splittings (i.e., lifting some pairs of incident edges and removing the vertex) and edge removals. Equivalently, vertices of $H$ are mapped to distinct vertices of $G$ (branch vertices), and edges of $H$ are mapped to pairwise edge-disjoint paths in $G$, each of them joining the branch vertices corresponding to the ends of the edge and not containing any other branch vertices. We show that there exists a function $d\colon N\to N$ such that for all graphs $H$ and $G$, if $G$ contains a strong immersion of the star $K_{1,d(\Delta(H))|V(H)|}$ whose branch vertices are $\Delta(H)$-edge-connected to one another, then $H$ is strongly immersed in $G$. This has a number of structural consequences for graphs avoiding a strong immersion of $H$. In particular, a class $\mathcal{G}$ of simple 4-edge-connected graphs contains all graphs of maximum degree 4 as strong immersions if and only if $\mathcal{G}$ has either unbounded maximum degree or unbounded tree-width.

LOCAL SPLITTING PROPERTIES OF ENDOMORPHISM RINGS OF PROJECTIVE MODULES

This paper deals with the unit groups of the endomor- phism rings of projective modules over polynomial rings and further over formal power series rings. A normal subgroup of the unit group is dened and discussed. The local splitting properties of elements of endomorphism rings of projective modules over polynomial rings are given. So, if P is a projective module over a ring A, then (id+ xEndA(x)(P(x))) is constructed. This is a normal subgroup of the unit group of endomorphism rings of a projective module over a poly- nomial ring. If (A,m) is a local ring with dim(A) ≥ µ(m), then we show that (id+ xEndA(x)(P(x))) is a normal subgroup of SLA(x)(P(x)). Also, we will show that under these conditions the similar conclusion can be drawn for the ring of formal power series. In section 3, let P be a projective A-module. Let s1, s2 ∈ A be such that As1 + As2 = A. Then we use the splitting lemma of Quillen to show that every element of (idPs1s2 + xEnd As1s2(x) (Ps1s2 (x))) has two decompositions. Finally, we generalize the result to Theorem 3.4.

A numerical approach to some basic theorems in singularity theory

In this paper, we give the explicit bounds for the data of objects involved in some basic theorems of singularity theory: the inverse, implicit and rank theorems for Lipschitz mappings, the splitting lemma and the Morse lemma, the density and openness of Morse functions. We expect that the results will make singularities more applicable and will be useful for numerical analysis and some fields of computing.

Asymmetric hydrogenation catalysis via ion-pairing in chiral ionic liquids

The relevant interactions between the organic ions of ionic liquids (ILs) include van der Waals forces, hydrogen bonding and coulombic interactions. In solutions these interactions lead to the formation of aggregates. Depending on the strength of the interactions and the concentration of the IL in the solvent, different aggregates are formed with ion-pairs being the dominant species at low concentrations. A probe for investigating these interactions is the enantioselectivity of a reaction carried out on a prochiral ion in presence of its chiral counter-ion. As asymmetric induction requires structurally well-defined interactions, the induced enantioselectivity can be correlated to ion aggregate formation. The first proof of this concept was provided by the asymmetric hydrogenation of [N-(3′-oxobutyl)-N-methylimidazolium]-[(R)-camphorsulfonate] and -[(S)-camphorsulfonate] using a heterogeneous Ru on C contact. Depending on the substrate IL concentration an enantioselectivity of up to 94% could be realized. The catalytic results were linked to the probability of the ion pair formation as evidenced by DOSY-NMR measurements and dielectric relaxation spectroscopy. The expansion of this principle to chiral ionic liquids that can be easily decomposed to chiral building blocks of synthetic relevance led us to the development of two approaches: a) the application of an IL substrate consisting of a chiral cation and a keto-functionalized, prochiral carboxylate counter-ion allowing for the synthesis of chiral esters in the sequence of IL hydrogenation and esterification; and b) reversible binding of a prochiral substrate to the cation of an IL with a chiral, enantiopure anion allowing the synthesis of neutral chiral building blocks in a sequence of binding, hydrogenation and splitting. The latter approach was tested for two ILs based on the hydroxyl-functionalized cation cholinium to which a prochiral keto functionalized carboxylic acids was reversibly attached via esterification. The hydrogenation of such prochiral ester cation in the presence of enantiopure anions resulted in ees of up to 63%.

Skew-commutator relations and Gröbner-Shirshov basis of quantum group of type F 4

We give a Grobner-Shirshov basis of quantum group of type F4 by using the Ringel-Hall algebra approach. We compute all skew-commutator relations between the isoclasses of indecomposable representations of Ringel-Hall algebras of type F4 by using an ‘inductive’ method. Precisely, we do not use the traditional way of computing the skew-commutative relations, that is first compute all Hall polynomials then compute the corresponding skew-commutator relations; instead, we compute the ‘easier’ skew-commutator relations which correspond to those exact sequences with middle term indecomposable or the split exact sequences first, then ‘deduce’ others from these ‘easier’ ones and this in turn gives Hall polynomials as a byproduct. Then using the composition-diamond lemma prove that the set of these relations constitute a minimal Grobner-Shirshov basis of the positive part of the quantum group of type F4. Dually, we get a Grobner-Shirshov basis of the negative part of the quantum group of type F4. And finally, we give a Grobner-Shirshov basis for the whole quantum group of type F4.

On the number of terms in the middle of almost split sequences over cycle-finite artin algebras

AbstractWe prove that the number of terms in the middle of an almost split sequence in the module category of a cycle-finite artin algebra is bounded by 5.

On Exact Sequence of Semimodules over Semirings

We introduce the notion of exact sequence of semimodules over semirings using maximal homomorphisms and generalize some results of module theory to semimodules over semirings. Indeed, we prove, “If is a split exact sequence of -semimodules and maximal -semimodule homomorphisms with is a strong subsemimodule of , then ”. Also, some results of commutative diagram of -semimodules and maximal -homomorphisms having exact rows are obtained.

Ideal approximation theory

Let (A;E) be an exact category and F⊆Ext a subfunctor. A morphism φ in A is an F-phantom if the pullback of an E-conflation along φ is a conflation in F. If the exact category (A;E) has enough injective objects and projective morphisms, it is proved that an ideal I of A is special precovering if and only if there is a subfunctor F⊆Ext with enough injective morphisms such that I is the ideal of F-phantom morphisms. A crucial step in the proof is a generalization of Salce’s Lemma for ideal cotorsion pairs: if I is a special precovering ideal, then the ideal cotorsion pair (I,I⊥) generated by I in (A;E) is complete. This theorem is used to verify: (1) that the ideal cotorsion pair cogenerated by the pure-injective modules of R-Mod is complete; (2) that the ideal cotorsion pair cogenerated by the contractible complexes in the category of complexes Ch(R-Mod) is complete; and, using Auslander and Reiten’s theory of almost split sequences, (3) that the ideal cotorsion pair cogenerated by the Jacobson radical Jac(Λ-mod) of the category Λ-mod of finitely generated representations of an Artin algebra is complete.

A simple method for obtaining the maximal correlation coefficient and related characterizations

We provide a method that enables the simple calculation of the maximal correlation coefficient of a bivariate distribution, under suitable conditions. In particular, the method readily applies to known results on order statistics and records. As an application we provide a new characterization of the exponential distribution: Under a splitting model on independent identically distributed observations, it is the (unique, up to a location-scale transformation) parent distribution that maximizes the correlation coefficient between the records among two different branches of the splitting sequence.

EXISTENCE OF ALMOST SPLIT SEQUENCES VIA REGULAR SEQUENCES

AbstractLet $(R, \mathfrak{m})$ be a Cohen–Macaulay complete local ring. We will apply an inductive argument to show that for every nonprojective locally projective maximal Cohen–Macaulay object $ \mathcal{X} $ of the morphism category of $R$ with local endomorphism ring, there exists an almost split sequence ending in $ \mathcal{X} $. Regular sequences are exploited to reduce the Krull dimension of $R$ on which the inductive argument is established. Moreover, the Auslander–Reiten translate of certain objects is described.

Auslander-Reiten components for some Knörr lattices

The almost split sequences terminating in certain Knorr lattices for finite groups are shown to have indecomposable middle terms. The tree classes of the connected components of the stable Auslander-Reiten quiver containing certain Knorr lattices for elementary abelian $p$-groups are shown to be $A_\infty.$