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.     

Bounds for Arithmetic Degrees

ABSTRACT

Let I be a monomial ideal with minimal monomial generators m₁,…, m_s, and assume that deg(m₁) ≥deg(m₂) ≥ … ≥deg(m_s). Among other things, we prove that the arithmetic degree of I is bounded above by deg(m₁)…deg(m_mht(I)), where mht(I) is the maximal height of associated primes of I. This bound is shaper than the one given in [12 Sturmfels, B., Trung, N. V., Vogel, W. (1995). Bounds on degrees of projective schemes. Math. Ann. 302:417–432.[Crossref], [Web of Science ®] , [Google Scholar]] and more natural than the one given in [9 Hoa, L. T., Trung, N. V. (1998). On the Castelnuovo-Mumford regularity and the arithmetic degree of monomial ideals. Math. Z. 229:519–537.[Crossref], [Web of Science ®] , [Google Scholar]]. In addition, we point out that adeg(I) ≠ adeg(Gin(I)) in general and conjecture that adeg(I) = adeg(Gin(I)) if and only if R/I is sequentially Cohen–Macaulay.     

Central Extensions and Derivations of the Lie Algebras of Skew Derivations for the Quantum Torus

ABSTRACT

In this article, we study the central extensions and derivations of the Lie algebra of skew derivations for the quantum torus. The results of the article generalize those obtained in Jiang and Meng (1998a Jiang , C. , Meng , D. ( 1998a ). The derivation algebra of the associative algebra C _q [X, Y, X ^?1, Y ^?1] . Comm. Algebra 26 : 1723 – 1736 . [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar] b Jiang , C. , Meng , D. ( 1998b ). The automorphism group of the derivation algebra of the Virasoro-like algebra . Adv. Math. (China) 27 : 175 – 183 . [CSA]  [Google Scholar]) and Kirkman et al. (1994 Kirkman , E. , Procesi , C. , Small , L. ( 1994 ). A q-analog for the Virasoro algebra . Comm. Algebra 22 : 3755 – 3774 . [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]).     

Heart of Irreducible Morphisms of Bounded Complexes

In [9 Giraldo , H. , Merklen , H. ( 2009 ). Irreducible morphisms of categories of complexes . Journal of Algebra 321 : 2716 – 2736 .[Crossref], [Web of Science ®] , [Google Scholar]] Giraldo and Merklen studied irreducible morphisms in the categories 𝒞(𝒜) and D^?(Λ), where 𝒞(𝒜) is the category of complexes over an abelian Krull–Schmidt category 𝒜 and D^?(Λ) is the derived category of the bounded above complexes of finite generate left modules, over an Artin algebra Λ. In this work, we continue the study of irreducible morphism having one finite irreducible truncation.     

Lifting of polynomial symplectomorphisms and deformation quantization

We study the problem of lifting of polynomial symplectomorphisms in characteristic zero to automorphisms of the Weyl algebra by means of approximation by tame automorphisms. In 1983, Anick proved the fundamental result on approximation of polynomial automorphisms. We obtain similar approximation theorems for symplectomorphisms and Weyl algebra authomorphisms. We then formulate the lifting problem. More precisely, we prove the possibility of lifting of a symplectomorphism to an automorphism of the power series completion of the Weyl algebra of the corresponding rank. The lifting problem has its origins in the context of deformation quantization of the a?ne space and is closely related to several major open problems in algebraic geometry and ring theory.

This paper is a continuation of the study [19 Kanel Belov, A., Razavinia, F., Zhang, W. (2017). Bergman’s centralizer theorem and quantization. Commun. Algebra 1–7.[Taylor &; Francis Online] , [Google Scholar]].     

Une Neutralisation Explicite de L'algèbre de Weyl Quantique Complétée

Soit p un nombre premier. Nous établissons l'existence de neutralisations de divers complétés de l'algèbre de Weyl quantique spécialisée en une racine de l'unité primitive d'ordre p (qui est “génériquement” une algèbre d'Azumaya) et donnons en particulier un énoncé de neutralisation explicite relevant celui construit en caractéristique p dans [3 Gros , M. , Le Stum , B. , Quiros , A. ( 2010 ). A Simpson correspondence in positive characteristic . Publ. RIMS Kyoto Univ. 46 : 1 – 35 .[Crossref], [Web of Science ®] , [Google Scholar]].

Let p be a prime number. We establish the existence of neutralizations of various completions of the quantum Weyl algebra specialized at a primitive root of unity of prime order p (which is “generically” an Azumaya algebra) and, in particular, we give a statement of explicit neutralization similar to the one built in characteristic p in [3 Gros , M. , Le Stum , B. , Quiros , A. ( 2010 ). A Simpson correspondence in positive characteristic . Publ. RIMS Kyoto Univ. 46 : 1 – 35 .[Crossref], [Web of Science ®] , [Google Scholar]].     

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.     

Corrigendum

We correct Proposition 4.21 in Roch et al., “A Sequence Algebra of Finite Sections Convolution and Multiplication” [2 S. Roch , P. A. Santos , and B. Silbermann ( 2010 ). A sequence algebra of finite sections, convolution and multiplication operators on L ^p (?) . Numer. Funct. Anal. Optimiz. 31 : 45 – 47 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] and its consequences. The correction amounts to the observation that the spectrum of X is a single circular arc, not a lens.     

Asymptotic Growth of Associated Primes of Certain Graph Ideals

We specify a class of graphs, H _t , and characterize the irreducible decompositions of all powers of the cover ideals. This gives insight into the structure and stabilization of the corresponding associated primes; specifically, providing an answer to the question “For each integer t ≥ 0, does there exist a (hyper) graph H _t such that stabilization of associated primes occurs at n ≥ (χ(H _t ) ?1) + t?” [4 Francisco , C. A. , Hà , H. T. , Van Tuyl , A. ( 2011 ). Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals . J. Algebra 331 : 224 – 242 .[Crossref], [Web of Science ®] , [Google Scholar]]. For each t, H _t has chromatic number 3 and associated primes that stabilize at n = 2 + t.     

Non–existence of theta–shaped self–similarly shrinking networks moving by curvature

We prove that there are no networks homeomorphic to the Greek “Theta” letter (a double cell) embedded in the plane with two triple junctions with angles of 120 degrees, such that under the motion by curvature they are self–similarly shrinking.

This fact completes the classification of the self–similarly shrinking networks in the plane with at most two triple junctions, see [5 Chen, X., Guo, J.-S. (2007). Self-similar solutions of a 2-D multiple-phase curvature flow. Phys. D. 229(1):22–34.[Crossref], [Web of Science ®] , [Google Scholar], 10 Hättenschweiler, J. (2007). Mean curvature flow of networks with triple junctions in the plane. Master’s thesis. ETH Zürich. [Google Scholar], 25 Schnürer, O. C., Azouani, A., Georgi, M., Hell, J., Nihar, J., Koeller, A., Marxen, T., Ritthaler, S., Sáez, M., Schulze, F., Smith, B. (2011). Evolution of convex lens–shaped networks under the curve shortening flow. Trans. Am. Math. Soc. 363(5):2265–2294.[Crossref], [Web of Science ®] , [Google Scholar], 2 Baldi, P., Haus, E., Mantegazza, C. (2016). Networks self-similarly moving by curvature with two triple junctions. Networks self-similarly moving by curvature with two triple junctions. 28(2017):323–338. [Google Scholar]].     

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]).     

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.     

Weyl Modules Associated to Kac–Moody Lie Algebras

Local Weyl modules were originally defined for affine Lie algebras by Chari and Pressley in [5 Chari, V., Pressley, A. (2001). Weyl modules for classical and quantum affine algebras. Represent. Theory 5:191–223 (electronic).[Crossref] , [Google Scholar]]. In this paper we extend the notion of local Weyl modules for a Lie algebra 𝔤 ?A, where 𝔤 is any Kac–Moody algebra and A is any finitely generated commutative associative algebra with unit over ?, and prove a tensor product decomposition theorem which generalizes result in [2 Chari, V., Fourier, G., Khandai, T. (2010). A categorical approach to Weyl modules. Transform. Groups 15(3):517–549.[Crossref], [Web of Science ®] , [Google Scholar], 5 Chari, V., Pressley, A. (2001). Weyl modules for classical and quantum affine algebras. Represent. Theory 5:191–223 (electronic).[Crossref] , [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.     

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.     

Algebraic Entropy of Generalized Shifts on Direct Products

For a set Γ, a function λ: Γ → Γ and a nontrivial abelian group K, the \emphgeneralized shift σ_λ: K ^Γ → K ^Γ is defined by (x _i ) _i∈Γ ? (x _λ(i)) _i∈Γ [3 Ayatollah Zadeh Shirazi , F. , Heidari Ardi , F. , Karami Kabir , N. ( 2008 ). A note on shift theory . Math. Pannon. 19 : 187 – 195 . [Google Scholar]]. In this article we compute the algebraic entropy of σ_λ; it is either zero or infinite, depending exclusively on the properties of λ. This solves two problems posed in [2 Akhavin , M. , Ayatollah Zadeh Shirazi , F. , Dikranjan , D. , Giordano Bruno , A. , Hosseini , A. ( 2009 ). Algebraic entropy of shift endomorphisms on abelian groups . Quaest. Math. 32 : 529 – 550 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Classical Morita equivalence and linear topologies; applications to quasi-dualities

ABSTRACT

In this note,we answer a question of Hong et al. (2003 Hong , C. Y. , Kim , N. K. , Kwak , T. K. ( 2003 ). On skew Armendariz rings . Comm. Alg. 31 ( 1 ): 103 – 122 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) by proving that if α is a monomorphism of a reduced ring R, and R is α-skew Armendariz, then R is α-rigid.     

Subgroup Depth and Twisted Coefficients

Danz computes the depth of certain twisted group algebra extensions in [10 Danz, S. (2011). The depth of some twisted group extensions. Comm. Alg. 39:1–15. [Google Scholar]], which are less than the values of the depths of the corresponding untwisted group algebra extensions in [8 Burciu, S., Kadison, L., Külshammer, B. (2011). On subgroup depth (with an appendix by B. Külshammer and S. Danz). I. E. J. A. 9:133–166. [Google Scholar]]. In this article, we show that the closely related h-depth of any group crossed product algebra extension is less than or equal to the h-depth of the corresponding (finite rank) group algebra extension. A convenient theoretical underpinning to do so is provided by the entwining structure of a right H-comodule algebra A and a right H-module coalgebra C for a Hopf algebra H. Then A ? C is an A-coring, where corings have a notion of depth extending h-depth. This coring is Galois in certain cases where C is the quotient module Q of a coideal subalgebra R ? H. We note that this applies for the group crossed product algebra extension, so that the depth of this Galois coring is less than the h-depth of H in G. Along the way, we show that the subgroup depth behaves exactly like the combinatorial depth with respect to the core of a subgroup, and extend results in [22 Kadison, L. (2014). Hopf subalgebras and tensor powers of generalized permutation modules. J. Pure Appl. Alg. 218:367–380.[Crossref], [Web of Science ®] , [Google Scholar]] to coideal subalgebras of finite dimension.     

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]].     

Dominance and Transmissions in Supertropical Valuation Theory

This article is a sequel of [4 Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2011 ). Supertropical semirings and supervaluations . J. Pure and Applied Alg. 215 ( 10 ): 2431 – 2463 .[Crossref], [Web of Science ®] , [Google Scholar]], where we defined supervaluations on a commutative semiring R and studied a dominance relation ? ≥ ψ between supervaluations ? and ψ on R, aiming at an enrichment of the algebraic tool box for use in tropical geometry.

A supervaluation ?: R → U is a multiplicative map from R to a supertropical semiring U, cf. [4 Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2011 ). Supertropical semirings and supervaluations . J. Pure and Applied Alg. 215 ( 10 ): 2431 – 2463 .[Crossref], [Web of Science ®] , [Google Scholar]], [7 Izhakian , Z. , Rowen , L. ( 2011 ). Supertropical matrix algebra . Israel J. Math. 182 ( 1 ): 383 – 424 .[Crossref], [Web of Science ®] , [Google Scholar]], [8 Izhakian , Z. , Rowen , L. ( 2010 ). Supertropical polynomials and resultants . J. Alg. 324 : 1860 – 1886 . (Preprint at arXiv:0902.2155.) [Crossref], [Web of Science ®] , [Google Scholar]], [5 Izhakian , Z. , Knebusch , M. , Rowen , L. Supertropical monoids: Basics and canonical factorization . Preprint at arXiv:1108.1880 . [Google Scholar]], [9 Maclane , S. ( 1998 ). Categories for the Working Mathemtician. , 4th ed. Springer Vereag . [Google Scholar]], with further properties, which mean that ? is a sort of refinement, or covering, of an m-valuation (= monoid valuation) v: R → M. In the most important case, that R is a ring, m-valuations constitute a mild generalization of valuations in the sense of Bourbaki [1 Bourbaki , N. Algèbre Commutative VI, §3 No. 1 . [Google Scholar]], while ? ≥ ψ means that ψ: R → V is a sort of coarsening of the supervaluation ?. If ?(R) generates the semiring U, then ? ≥ ψ iff there exists a “transmission” α: U → V with ψ = α ○ ?.

Transmissions are multiplicative maps with further properties, cf. [4 Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2011 ). Supertropical semirings and supervaluations . J. Pure and Applied Alg. 215 ( 10 ): 2431 – 2463 .[Crossref], [Web of Science ®] , [Google Scholar], Section 5]. Every semiring homomorphism α: U → V is a transmission, but there are others which lack additivity, and this causes a major difficulty. In the main body of the article we study surjective transmissions via equivalence relations on supertropical semirings. We put special emphasis on homomorphic equivalence relations. Even those are often much more complicated than congruences by ideals in usual commutative algebra.