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.     

Loewy Modules with Finite Loewy Invariants and Max Modules with Finite Radical Invariants

Over a commutative ring R, a module is artinian if and only if it is a Loewy module with finite Loewy invariants [5 Facchini , A. ( 1981 ). Loewy and artinian modules over commutative rings . Ann. Mat. Pura Appl. 128 : 359 – 374 .[Crossref], [Web of Science ®] , [Google Scholar]]. In this paper, we show that this is not necesarily true for modules over noncommutative rings R, though every artinian module is always a Loewy module with finite Loewy invariants. We prove that every Loewy module with finite Loewy invariants has a semilocal endomorphism ring, thus generalizing a result proved by Camps and Dicks for artinian modules [3 Camps , R. , Dicks , W. ( 1993 ). On semilocal rings . Israel J. Math. 81 : 203 – 211 .[Crossref], [Web of Science ®] , [Google Scholar]]. Finally, we obtain similar results for the dual class of max modules.     

A Braided Version of Some Results of Skryabin

We extend the main result of Skryabin in [9 Skryabin , S. ( 2007 ). Projectivity and freeness over comodule algebras . Trans. Amer. Math. Soc. 359 : 2597 – 2623 .[Crossref], [Web of Science ®] , [Google Scholar]] to Yetter-Drinfeld Hopf algebras over an arbitrary Hopf algebra H with bijective antipode.     

Quasi-Deformations of 𝔰𝔩2(𝔽) Using Twisted Derivations

In this article we apply a method devised in Hartwig, Larsson, and Silvestrov (2006 Hartwig , J. T. , Larsson , D. , Silvestrov , S. D. ( 2006 ). Deformations of Lie algebras using σ-derivations . J. Algebra 295 : 314 – 361 .[Crossref], [Web of Science ®] , [Google Scholar]) and Larsson and Silvestrov (2005a Larsson , D. , Silvestrov , S. D. (2005a). Quasi-hom-Lie algebras, Central extensions and 2-cocycle-like identities. J. Algebra 288:321–344.[Crossref], [Web of Science ®] , [Google Scholar]) to the simple 3-dimensional Lie algebra 𝔰𝔩₂(𝔽). One of the main points of this deformation method is that the deformed algebra comes endowed with a canonical twisted Jacobi identity. We show in the present article that when our deformation scheme is applied to 𝔰𝔩₂(𝔽) we can, by choosing parameters suitably, deform 𝔰𝔩₂(𝔽) into the Heisenberg Lie algebra and some other 3-dimensional Lie algebras in addition to more exotic types of algebras, this being in stark contrast to the classical deformation schemes where 𝔰𝔩₂(𝔽) is rigid.     

The infinitesimal and the little q-Schur superalgebras

In this paper, based on the results in [8 Du, J., Gu, H.-X. (2014). A realization of the quantum supergroup U(𝔤𝔩_m|n). J. Algebra 404:60–99.[Web of Science ®] , [Google Scholar]] we give a monomial basis for q-Schur superalgebra and then a presentation for it. The presentation is different from that in [12 El Turkey, H., Kujawa, J. (2012). Presenting Schur superalgebras. Pacific J. Math., 262(2):285–316.[Crossref], [Web of Science ®] , [Google Scholar]]. Imitating [3 Cox, A. G. (1997). On some applications of infinitesimal methods to quantum groups and related algebras. Ph.D. Thesis. University of London. [Google Scholar]] and [7 Du, J., Fu, Q., Wang, J.-P. (2005). Infinitesimal quantum 𝔤𝔩_n and little q-Schur algebras. J. Algebra 287:199–233.[Crossref], [Web of Science ®] , [Google Scholar]], we define the infinitesimal and the little q-Schur superalgebras. We give a “weight idempotent presentation” for infinitesimal q-Schur superalgebras. The BLM bases and monomial bases of little q-Schur superalgebras are obtained, and dimension formulas of infinitesimal and little q-Schur superalgebras are deduced.     

Bialgebra Cohomology of the Duals of a Class of Generalized Taft Algebras

We compute explicitly the bialgebra cohomology of the duals of the generalized Taft algebras, which are noncommutative, noncocommutative finite-dimensional Hopf algebras. In order to do this, we use an identification of this cohomology with an Ext algebra (Taillefer, 2004a Taillefer , R. ( 2004a ). Cohomology theories of Hopf bimodules and cup-product . Alg. and Representation Theory 7 : 471 – 490 . [Google Scholar]) and a result describing the Drinfeld double of the dual of a generalized Taft algebra up to Morita equivalence (Erdmann et al., 2006 Erdmann , K. , Green , E. L. , Snashall , N. , Taillefer , R. ( 2006 ). Representation theory of the Drinfeld doubles of a family of Hopf algebras . J. Pure and Applied Algebra 204 ( 2 ): 413 – 454 .[Crossref], [Web of Science ®] , [Google Scholar]).     

Larson–Sweedler Theorem and Some Properties of Discrete Type in (G-Cograded) Multiplier Hopf Algebras

We extend the Larson–Sweedler theorem to group-cograded multiplier Hopf algebras introduced in Abd El-hafez et al. (2004 Abd El-hafez , A. T. , Delvaux , L. , Van Daele , A. ( 2004 ). Group-cograded multiplier Hopf (?-)algebra. Math. QA/0404026 . To appear in Algebras and Representation Theory . [CSA]  [Google Scholar]), by showing that a group-cograded multiplier bialgebra with finite-dimensional unital components is a group-cograded multiplier Hopf algebra if and only if it possesses a nondegenerate left cointegral. We also generalize the theory of multiplier Hopf algebras of discrete type in Van Daele and Zhang (1999 Van Daele , A. , Zhang , Y. ( 1999 ). Multiplier Hopf algebras of discrete type . J. Algebra 214 : 400 – 417 . [CSA] [CROSSREF]  [Google Scholar]) to group-cograded multiplier Hopf algebras. Our results are applicable to Hopf group-coalgebras in the sense of Turaev (2000 Turaev , V. G. ( 2000 ). Homotopy field theory in dimension 3 and crossed group-categories . Preprint GT/0005291. [CSA]  [Google Scholar]). Finally, we study regular multiplier Hopf algebras of η -discrete type.     

Construction of CPs-Stratified Algebras

The results of [7 Dlab , V. , Ringel , C. M. ( 1992 ). The module theoretical approach to quasi-hereditary algebras. In: Tachikawa, H., Brenner, S. eds. Representations of Algebras and Related Topics, London Math. Society Lecture Note Series 168:200–224 . [Google Scholar]] and [2 Ágoston , I. , Dlab , V. , Lukács , E. ( 2011 ). Constructions of stratified algebras . Comm. Algebra 39 : 2545 – 2553 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] gave a recursive construction for all quasi-hereditary and standardly stratified algebras starting with local algebras and suitable bimodules. Using the notion of stratifying pairs of subcategories, introduced in [3 Ágoston , I. , Lukács , E. Stratifying pairs of subcategories for CPS-stratified algebras . To appear in Journal of Algebra and Its Applications , p. 11 . [Google Scholar]], we generalize these earlier results to construct recursively all CPS-stratified algebras.     

Conjugacy Classes of P-Cycles of Type D in Alternating Groups

We classify the conjugacy classes of p-cycles of type D in alternating groups. This finishes the open cases in [3 Andruskiewitsch , N. , Fantino , F. , Graña , M. , Vendramin , L. ( 2011 ). Finite-dimensional pointed Hopf algebras with alternating groups are trivial . Ann. Mat. Pura Appl 190 : 225 – 245 .[Web of Science ®] , [Google Scholar]]. Also we determine all the subracks of those conjugacy classes which are not of type D.     

The minimal non-Koszul A(Γ)

The algebras A(Γ), where Γ is a directed layered graph, were first constructed by Gelfand et al. [5 Gelfand, I., Serconek, S., Retakh, V., Wilson, R. L. (2005). On a class of algebras associated to directed graphs. Selecta Math. (N.S.) 11(2):281–295.[Crossref], [Web of Science ®] , [Google Scholar]]. These algebras are generalizations of the algebras Q_n, which are related to factorizations of non-commutative polynomials. It was originally conjectured that these algebras were Koszul. In 2008, Cassidy and Shelton found a counterexample to this claim, a non-Koszul A(Γ) corresponding to a graph Γ with 18 edges and 11 vertices. We produce an example of a directed layered graph Γ with 13 edges and 9 vertices, which produces a non-Koszul A(Γ). We also show this is the minimal example with this property.     

Classification of Pointed Fusion Categories of dimension 8 up to weak Morita equivalence

In this paper we give a complete classification of pointed fusion categories over ? of global dimension 8. We first classify the equivalence classes of pointed fusion categories of dimension 8, and then we proceed to determine which of these equivalence classes have equivalent categories of modules following the procedure presented in [9 Naidu, D. (2007). Categorical Morita equivalence for group-theoretical categories. Commun. Algebra 35(11):3544–3565.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], 11 Uribe, B. (2017). On the classification of pointed fusion categories up to weak Morita equivalence. Pac. J. Math. 290(2):437–466.[Crossref], [Web of Science ®] , [Google Scholar]]. The results of this paper permit to recover the classification of twisted quantum doubles of groups of order 8 up to gauge equivalence of braided quasi-Hopf algebras that was previously done in [6 Mason, C., Ng, S.-H (2001). Group cohomology and gauge equivalence of some twisted quantum doubles. Trans. Am. Math. Soc. 353(9):3465–3509.[Crossref], [Web of Science ®] , [Google Scholar]] and [5 Goff, C., Mason, G., Ng, S.-H (2007). On the gauge equivalence of twisted quantum doubles of elementary abelian and extra-special 2-groups. J. Algebra 312(2):849–875.[Crossref], [Web of Science ®] , [Google Scholar]].     

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.     

Semilocal Convergence of Steffensen-Type Algorithms for Solving Nonlinear Equations

In this article, we provide a semilocal analysis for the Steffensen-type method (STTM) for solving nonlinear equations in a Banach space setting using recurrence relations. Numerical examples to validate our main results are also provided in this study to show that STTM is faster than other methods ([7 I. K. Argyros , J. Ezquerro , J. M. Gutiérrez , M. Hernández , and S. Hilout ( 2011 ). On the semilocal convergence of efficient Chebyshev-Secant-type methods . J. Comput. Appl. Math. 235 : 3195 – 3206 .[Crossref], [Web of Science ®] , [Google Scholar], 13 J. A. Ezquerro and M. A. Hernández ( 2009 ). An optimization of Chebyshev's method . J. Complexity 25 : 343 – 361 .[Crossref], [Web of Science ®] , [Google Scholar]]) using similar convergence conditions.     

Categorical Realizations of Quivers

We introduce and study categorical realizations of quivers. This construction generalizes comma categories and includes representations of quivers on categories, twisted representations of quivers (in the sense of [9 Gothen, P. B., King, Alastair D. (2005). Homological algebra of twisted quiver bundles. J. London Math. Soc. (2) 71(1):85–99.[Crossref], [Web of Science ®] , [Google Scholar]]), and bilinear pairings as special cases. In this general context, we prove the cancelation from direct sums, and show the existence and uniqueness of decomposition into a sum of indecomposable objects, provided certain assumptions hold. This yields similar results for the special cases just mentioned. Using similar ideas, we also prove a version of Fitting's Lemma for natural transformations between functors.     

Simple Modules and Primitive Ideals of Non-Noetherian Generalized Down-Up Algebras

Generalized down-up algebras were first introduced in Cassidy and Shelton (2004 Cassidy , T. , Shelton , B. ( 2004 ). Basic properties of generalized down-up algebras . J. Algebra 279 : 402 – 421 .[Crossref], [Web of Science ®] , [Google Scholar]). Their simple weight modules were classified in Cassidy and Shelton (2004 Cassidy , T. , Shelton , B. ( 2004 ). Basic properties of generalized down-up algebras . J. Algebra 279 : 402 – 421 .[Crossref], [Web of Science ®] , [Google Scholar]) in the noetherian case, and in Praton (2007 Praton , I. ( 2007 ). Simple weight modules of non-noetherian generalized down-up algebras . Comm. Algebra 35 : 325 – 337 .[Taylor &; Francis Online] , [Google Scholar]) in the non-noetherian case. Here we concentrate on non-noetherian down-up algebras. We show that almost all simple modules are weight modules. We also classify the corresponding primitive ideals.     

On the Twisting and Drinfel'd Double for Multiplier Hopf Algebras

We define the concept of “semiprime” for preradicals and for submodules, and we prove some properties that relate both of them. Related concepts are defined in article by Bican et al. [2 Bican , L. , Jambor , P. , Kepka , T. , Nemec , P. ( 1980 ). Prime and coprime modules . Fundamenta Mathematicae CVII , 33 – 45 . [Google Scholar]] and by Van den Berg and Wisbauer [9 Van den Berg , J. , Wisbauer , R. ( 2001 ). Duprime and dusemiprime modules . Journal of Pure and Applied Algebra 165 : 337 – 356 .[Crossref], [Web of Science ®] , [Google Scholar]]. For any ring, we compare the least semiprime preradical, the Jacobson radical and the join of all nilpotent preradicals, and we characterize V-rings in terms of these three preradicals. We study the least semiprime preradical above any preradical and we prove some of its properties. Using “Amitsur constructions” we define another related operators and prove some of their properties.     

On a Class of Finite-dimensional Semisimple Hopf Algebras

Let H be a finite-dimensional and semisimple Hopf algebra over an algebraically closed field of characteristic 0 such that H has exactly one isomorphism class of simple modules that have not dimension 1. These Hopf algebras were the object of study in, for instance, [1 Artamonov , V. A. ( 2007 ). Semisimple finite-dimensional Hopf algebras . Sbornik: Mathematics 198 ( 9 ): 1221 – 1245 .[Crossref], [Web of Science ®] , [Google Scholar]] and [9 Mukhatov , R. B. ( 2009 ). On semisimple finite-dimensional Hopf algebras . Fundamentalnaya i Prikladnaya Matematika 15 ( 2 ): 133 – 143 . [Google Scholar]]. In this paper we study this property in the context of certain abelian extensions of group algebras and give a group theoretical criterion for such Hopf algebras to be of the above type. We also give a classification result in a special case thereof.     

Graded Left Symmetric Pseudo-algebras

In the present article, we introduce G-graded left symmetric H-pseudoalgebras, where G is a grading group, and H is a cocommutative Hopf algebra. Some results about associative H-pseudoalgebras in [23 Retakh , A. ( 2004 ). Unital associative pseudoalgebras and their representations . J. Algebra 227 : 769 – 805 .[Crossref] , [Google Scholar]] are generalized. The commutator algebras of the G-graded left symmetric H-pseudo-algebras are Lie H-pseudoalgebras, which are classified when the grading group is trivial in [3 Bakalov , B. , D'Andrea , A. , Kac , V. G. ( 2001 ). Theory of finite pseudoalgebras . Adv. in Math. 162 : 1 – 140 .[Crossref], [Web of Science ®] , [Google Scholar]]. We investigate the left symmetric structure of Lie H-pseudoalgebras W(𝔟), S(𝔟), and He defined in [3 Bakalov , B. , D'Andrea , A. , Kac , V. G. ( 2001 ). Theory of finite pseudoalgebras . Adv. in Math. 162 : 1 – 140 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Strongly J-Clean Matrices Over Local Rings

An element of a ring is called strongly J-clean provided that it can be written as the sum of an idempotent and an element in its Jacobson radical that commute. We investigate, in this article, a single strongly J-clean 2 × 2 matrix over a noncommutative local ring. The criteria on strong J-cleanness of 2 × 2 matrices in terms of a quadratic equation are given. These extend the corresponding results in [8 Li , B. ( 2009 ). Strongly clean matrix rings over noncommutative local rings . Bull. Korean Math. Soc. 46 : 71 – 78 .[Crossref], [Web of Science ®] , [Google Scholar], Theorems 2.7 and 3.2], [9 Li , Y. ( 2007 ). Strongly clean matrix rings over local rings . J. Algebra 312 : 397 – 404 .[Crossref], [Web of Science ®] , [Google Scholar], Theorem 2.6], and [11 Yang , X. , Zhou , Y. ( 2008 ). Strongly cleanness of the 2 × 2 matrix ring over a general local ring . J. Algebra 320 : 2280 – 2290 .[Crossref], [Web of Science ®] , [Google Scholar], Theorem 7].