Finitary Incidence Algebras

It is well known that incidence algebras can be defined only for locally finite partially ordered sets (Doubilet et al., 1972 Doubilet , P. , Rota , G.-C. , Stanley , R. ( 1972 ). On the foundations of combinatorial theory (VI): The idea of generating function . In: Proc. of the Sixth Berkely Symp. on Math. Stat. and Probab . v. II , Univ. of Calif. Press , pp. 267 – 318 . [Google Scholar]; Stanley 1986 Stanley , R. P. ( 1986 ). Enumerative Combinatorics . v. 1 . Monterey, CA : Wadsworth &; Brooks/Cole .[Crossref] , [Google Scholar]). At the same time, for example, the poset of cells of a noncompact cell partition of a topological space is not locally finite. On the other hand, some operations, such as the order sum and the order product (Stanley, 1986 Stanley , R. P. ( 1986 ). Enumerative Combinatorics . v. 1 . Monterey, CA : Wadsworth &; Brooks/Cole .[Crossref] , [Google Scholar]), do not save the locally finiteness. So it is natural to try to generalize the concept of incidence algebra.

In this article, we consider the functions in two variables on an arbitrary poset (finitary series), for which the convolution operation is defined. We obtain the generalization of incidence algebra—finitary incidence algebra and describe its properties: invertibility, the Jackobson radical, idempotents, regular elements. As a consequence a positive solution of the isomorphism problem for such algebras is obtained.     

Derived Brauer–Thrall Type Theorem I for Artin Algebras

We define the global cohomological range for artin algebras, and define the derived bounded algebras to be the algebras with finite global cohomological range, then we prove the first Brauer–Thrall type theorem for bounded derived categories of artin algebras, i.e., derived bounded algebras are precisely the derived finite algebras. Moreover, the main theorem establishes that the derived bounded artin algebras are just piecewise hereditary algebras of Dynkin type, and can be also characterized as those artin algebras with derived dimension zero, which can be regarded as a generalization of the results of Han–Zhang [11 Han, Y., Zhang, C. Brauer-Thrall type theorems for derived category, arXiv:1310.2777. [Google Scholar], Theorem 1] and Chen–Ye–Zhang [4 Chen, X. W., Ye, Y., Zhang, P. (2008). Algebras of derived dimension zero. Comm. Algebra 36:1–10.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], Theorem] in the context of finite-dimensional algebras over algebraically closed fields, respectively.     

Derived Equivalences Between Subrings

In this paper, we construct derived equivalences between two subrings of relevant Φ-Yoneda rings from an arbitrary short exact sequence in an abelian category. As a consequence, any short exact sequence in an abelian category gives rise to a derived equivalence between two subrings of endomorphism rings.     

Derived Equivalence Classification of Nonstandard Selfinjective Algebras of Domestic Type

We give a complete derived equivalence classification of all nonstandard representation-infinite domestic selfinjective algebras over an algebraically closed field. As a consequence, a complete stable equivalence classification of these algebras is obtained.     

On Derived Equivalences for Selfinjective Algebras

We show that if A is a representation-finite selfinjective Artin algebra, then every P ^? ? K ^b(𝒫 _A ) with Hom _K(Mod?A)(P ^?,P ^?[i]) = 0 for i ≠ 0 and add(P ^?) = add(νP ^?) is a direct summand of a tilting complex, and that if A, B are derived equivalent representation-finite selfinjective Artin algebras, then there exists a sequence of selfinjective Artin algebras A = B ₀, B ₁,…, B _m  = B such that, for any 0 ≤ i < m, B _i+1 is the endomorphism algebra of a tilting complex for B _i of length ≤ 1.     

Gorenstein injective envelopes and covers over n-perfect rings

We prove the existence of Gorenstein injective envelopes and covers over n-perfect rings for some classes of modules associated with a dualizing bimodule.     

Gorenstein injective envelopes and covers over two sided noetherian rings

We prove that the class of Gorenstein injective modules is both enveloping and covering over a two sided noetherian ring such that the character modules of Gorenstein injective modules are Gorenstein flat. In the second part of the paper we consider the connection between the Gorenstein injective modules and the strongly cotorsion modules. We prove that when the ring R is commutative noetherian of finite Krull dimension, the class of Gorenstein injective modules coincides with that of strongly cotorsion modules if and only if the ring R is in fact Gorenstein.     

Generalized Auslander-Reiten Conjecture and Derived Equivalences

In this note, we prove that the generalized Auslander-Reiten conjecture is preserved under derived equivalences between Artin algebras.     

Derived equivalences from mutations of quivers with potential

We show that Derksen–Weyman–Zelevinsky's mutations of quivers with potential yield equivalences of suitable 3-Calabi–Yau triangulated categories. Our approach is related to that of Iyama–Reiten and ‘Koszul dual’ to that of Kontsevich–Soibelman. It improves on previous work by Vitória. In Appendix A, the first-named author studies pseudo-compact derived categories of certain pseudo-compact dg algebras.     

Frobenius Extensions and Tilting Complexes

Let n ≥ 1 be an integer and π a permutation of I = {1, ⋯ ,n}. For any ring R, we provide a systematic construction of rings A which contain R as a subring and enjoy the following properties: (a) 1 = ∑  _i ∈ I e _i with the e _i orthogonal idempotents; (b) e _i x = xe _i for all i ∈ I and x ∈ R; (c) e _i A e _j  ≠ 0 for all i, j ∈ I; (d) e _i A _A  ≇ e _j A _A unless i = j; (e) every e _i Ae _i is a local ring whenever R is; (f) e _i A _A  ≅ Hom _R (Ae _π(i),R _R ) and _A Ae _π(i) ≅  _A Hom _R (e _i A, _R R) for all i ∈ I; and (g) there exists a ring automorphism η ∈ Aut(A) such that η(e _i ) = e _π(i) for all i ∈ I. Furthermore, for any nonempty π-stable subset J of I, the mapping cone of the multiplication map is a tilting complex. Dedicated to Takeshi Sumioka on the occasion of his 60th birthday.     

Coefficient characterization of linear differential equations with maximal symmetries

A characterization of the general linear equation in standard form admitting a maximal symmetry algebra is obtained in terms of a simple set of conditions relating the coefficients of the equation. As a consequence, it is shown that in its general form such an equation can be expressed in terms of only two arbitrary functions, and its connection with the Laguerre–Forsyth form is clarified. The characterizing conditions are also used to derive an infinite family of semi-invariants, each corresponding to an arbitrary order of the linear equation. Finally a simplifying ansatz is established, which allows an easier determination of the infinitesimal generators of the induced pseudo group of equivalence transformations, for all the three most common canonical forms of the equation.     

Amenable representations and finite injective von Neumann algebras

Let be the unitary group of a finite, injective von Neumann algebra . We observe that any subrepresentation of a group representation into is amenable in the sense of Bekka; this yields short proofs of two known results-one by Robertson, one by Haagerup-concerning group representations into .

     

Inverse functions of polynomials and orthogonal polynomials as operator monotone functions

We study the operator monotonicity of the inverse of every polynomial with a positive leading coefficient. Let be a sequence of orthonormal polynomials and the restriction of to , where is the maximum zero of . Then and the composite are operator monotone on . Furthermore, for every polynomial with a positive leading coefficient there is a real number so that the inverse function of defined on is semi-operator monotone, that is, for matrices , implies

     

On finitely injective modules and locally pure-injective modules over Prüfer domains

Over Matlis valuation domains there exist finitely injective modules which are not direct sums of injective modules, as well as complete locally pure-injective modules which are not the completion of a direct sum of pure-injective modules. Over Prüfer domains which are either almost maximal, or -local Matlis, finitely injective torsion modules and complete torsion-free locally pure-injective modules correspond to each other under the Matlis equivalence. Almost maximal Prüfer domains are characterized by the property that every torsion-free complete module is locally pure-injective. It is derived that semi-Dedekind domains are Dedekind.

     

Kaplansky Classes and Cotorsion Theories of Complexes

In this article we provide arguments for constructing Kaplansky classes in the category of complexes out of a Kaplansky class of modules. This leads to several complete cotorsion theories in such categories. Our method gives a unified proof for most of the known cotorsion theories in the category of complexes and can be applied to the category of quasi-coherent sheaves over a scheme as well as the category of the representations of a quiver.     

逆矩阵的判定及计算方法

线性代数是大学教育中一门难度较高的基础必修课程,而逆矩阵是教学过程中一个主要概念,对研究其他线性结构有着非常重要的作用.本文通过对逆矩阵定义的分析,汇总若干个判定矩阵是否可逆的方法,同时提供了多种逆矩阵的计算技巧,包括利用计算机技术简化繁琐的计算过程,这些都是学习者在学习过程中需要掌握的重要内容.本文旨在协助教师在开展教学时,能够举一反三,以点带面来引导学生将所学知识融合,注重知识点之间的相关性学习;同时,也帮助学习者能够更加全面的认识逆矩阵这一重要概念.     

Multicategories and varieties of many-sorted algebras

We give a direct proof of the fact that every variety of many-sorted universal algebras is rationally equivalent to a variety of algebras over some FSet-multicategory (a many-sorted analog of FSet-operads). The construction of this multicategory is straightforward and uses the free algebras of the given variety.     

Deformed Kazhdan-Lusztig elements and Macdonald polynomials

We introduce deformations of Kazhdan-Lusztig elements and specialised non-symmetric Macdonald polynomials, both of which form a distinguished basis of the polynomial representation of a maximal parabolic subalgebra of the Hecke algebra. We give explicit integral formula for these polynomials, and explicitly describe the transition matrices between classes of polynomials. We further develop a combinatorial interpretation of homogeneous evaluations using an expansion in terms of Schubert polynomials in the deformation parameters.