Root extraction in one-relator groups and slenderness

In this note we strengthen a result of Newman and use it to prove a conjecture of Nakamura stated in [10 Nakamura, J. (2015). Atomic properties of the Hawaiian earring group for HNN extensions. Commun. Algebra 43:4138–4147.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] that torsion-free one-relator groups are noncommutatively slender.     

Supercharacter Theory Constructions Corresponding to Schur Ring Products

Diaconis and Isaacs have defined the supercharacter theories of a finite group to be certain approximations to the ordinary character theory of the group [7 Diaconis , P. , Isaacs , I. M. ( 2008 ). Supercharacters and superclasses for algebra groups . Trans. Amer. Math. Soc. 360 : 2359 – 2392 .[Crossref], [Web of Science ®] , [Google Scholar]]. We make explicit the connection between supercharacter theories and Schur rings, and we provide supercharacter theory constructions which correspond to Schur ring products of Leung and Man [12 Leung , K. H. , Man , S. H. ( 1996 ). On Schur rings over cyclic groups, II . J. Algebra 183 : 273 – 285 .[Crossref], [Web of Science ®] , [Google Scholar]], Hirasaka and Muzychuk [10 Hirasaka , M. , Muzychuk , M. ( 2001 ). An elementary abelian group of rank 4 is a CI-group . J. Combin. Theory Ser. A 94 : 339 – 362 .[Crossref], [Web of Science ®] , [Google Scholar]], and Tamaschke [20 Tamaschke , O. ( 1970 ). On Schur-rings which define a proper character theory on finite groups . Math. Z. 117 : 340 – 360 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Dixmier's Problem 6 for the Weyl Algebra (The Generic Type Problem)

ABSTRACT

In Dixmier (1968 Dixmier , J. ( 1968 ). Sur les algèbres de Weyl . Bull. Soc. Math. France 96 : 209 – 242 . [CSA] [Crossref] , [Google Scholar]), the author posed six problems for the Weyl algebra A ₁ over a field K of characteristic zero. Problems 3, 6, and 5 were solved respectively by Joseph (1975 Joseph , A. ( 1975 ). The Weyl algebra—semisimple and nilpotent elements . Amer. J. Math. 97 ( 3 ): 597 – 615 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) and Bavula (2005a Bavula , V. V. ( 2005a ). Dixmier's Problem 5 for the Weyl algebra . J. Algebra 283 ( 2 ): 604 – 621 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]). Problems 1, 2, and 4 are still open. In this article a short proof is given to Dixmier's problem 6 for the ring of differential operators 𝒟 (X) on a smooth irreducible algebraic curve X. It is proven that, for a given maximal commutative subalgebra C of 𝒟 (X), (almost) all noncentral elements of it have the same type, more precisely, have exactly one of the following types: (i) strongly nilpotent; (ii) weakly nilpotent; (iii) generic; (iv) generic, except for a subset K*a + K of strongly semi-simple elements; (iv) generic, except for a subset K*a + K of weakly semi-simple elements, where K* := K\{0}. The same results are true for other popular algebras.     

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.     

On Multiplicative Invariants of Finite Reflection Groups

This article focuses on two recent results on multiplicative invariants of finite reflection groups: Lorenz (2001 Lorenz , M. ( 2001 ). Multiplicative invariants and semigroup algebras . Alg. and Rep. Theory 4 : 293 – 304 . [Google Scholar]) showed that such invariants are affine normal semigroup algebras, and Reichstein (2003 Reichstein , Z. ( 2003 ). SAGBI bases in rings of multiplicative invariants . Commentarii Math. Helvetici 78 ( 1 ): 185 – 202 .[Web of Science ®] , [Google Scholar]) proved that the invariants have a finite SAGBI basis. Reichstein (2003 Reichstein , Z. ( 2003 ). SAGBI bases in rings of multiplicative invariants . Commentarii Math. Helvetici 78 ( 1 ): 185 – 202 .[Web of Science ®] , [Google Scholar]) also showed that, conversely, if the multiplicative invariant algebra of a finite group G has a SAGBI basis, then G acts as a reflection group. There is no obvious connection between these two results. We will show that multiplicative invariants of finite reflection groups have a certain embedding property that implies both results simultaneously.     

Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator

In the very influential paper [4 Caffarelli, L.A., Silvestre, L. (2007). An extension problem related to the fractional Laplacian. Commun. Partial Differential Equations 32:1245–1260.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] Caffarelli and Silvestre studied regularity of (?Δ)^s, 0<s<1, by identifying fractional powers with a certain Dirichlet-to-Neumann operator. Stinga and Torrea [15 Stinga, P.R., Torrea, J. (2010). Extension problem and Harnack’s inequality for some fractional operators. Commun. Partial Differential Equations 35:2092–2122.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] and Galé et al. [7 Galé, J., Miana, P., Stinga, P.R. (2013). Extension problem and fractional operators: semigroups and wave equations. J. Evol. Eqn. 13:343–368.[Crossref], [Web of Science ®] , [Google Scholar]] gave several more abstract versions of this extension procedure. The purpose of this paper is to study precise regularity properties of the Dirichlet and the Neumann problem in Hilbert spaces. Then the Dirichlet-to-Neumann operator becomes an isomorphism between interpolation spaces and its part in the underlying Hilbert space is exactly the fractional power.     

Singular Solutions to the Quantum Yang–Baxter Equations

Let X and A be weak Hopf algebras in the sense of Li (1998 Li , F. ( 1998 ). Weak Hopf algebras and some new solutions of the quantum Yang–Baxter equation . J. Algebra 208 ( 1 ): 72 – 100 .[Crossref], [Web of Science ®] , [Google Scholar]). As in the case of Hopf algebras (Majid, 1990 Majid , S. ( 1990 ). Quasitriangular Hopf algebras and Yang–Baxter equations . Internat. J. Modern Phys. A 5 : 1 – 91 . [Google Scholar]), a weak bicrossed coproduct X∞ _R A is constructed by means of good regular R-matrices of the weak Hopf algebras X and A. Using this, we provide a new framework of obtaining singular solutions of the quantum Yang–Baxter equation by constructing weak quasitriangular structures over X∞ _R A when both X and A admit a weak quasitriangular structure. Finally, two explicit examples are given.     

Almost Dual Pairs and Definable Classes of Modules

In [7 Holm , H. ( 2010 ). Modules with cosupport and injective functors . Algebr. Represent. Theor. 13 : 543 – 560 .[Crossref], [Web of Science ®] , [Google Scholar]] Holm considers categories of right modules dual to those with support in a set of finitely presented modules. We extend some of his results by placing them in the context of elementary duality on definable subcategories. In doing so we also prove that dual modules have enough indecomposable direct summands.     

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.     

On Cohomological Periodicity After 2-Steps

The notion of cohomological periodicity after 1-step has been studied by Talelli in [7 Talelli , O. ( 1980 ) On cohomological periodicity for infinite groups . Comment. Math. Helvetici 55 : 178 – 192 .[Crossref], [Web of Science ®] , [Google Scholar], 8 Talelli , O. ( 1983 ). On groups with periodic cohomology after 1-step . Journal of Pure and Applied Algebra 30 : 85 – 93 .[Crossref], [Web of Science ®] , [Google Scholar]], and [9 Talelli , O. ( 1996 ). On cohomological periodicity isomorphisms . Bull. London Math. Soc. 28 : 600 – 602 .[Crossref], [Web of Science ®] , [Google Scholar]]. If a group G has periodic cohomology after 1-step, then G is the fundamental group of a graph of finite groups, which have periodic cohomology of the same period. Also, the fundamental group of a tree of finite groups, which have periodic cohomology of the same period, has periodic cohomology after 1-step. In this paper, we show that if a group G has only cyclic finite subgroups and is the fundamental group of a certain tree of groups, which have -steps.     

A Note on Maximal Properties of Some Subsemigroups of Finite Order-Preserving Transformation Semigroups

Both maximal idempotent-generated subsemigroups and maximal idempotent-generated regular subsemigroups of O _n were studied by Yang [10 Yang , X. , Lu , C. ( 2000 ). Maximal properties of some subsemigroups in finite order-preserving transformation semigroups . Communications in Algebra 28 : 3125 – 3135 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. The purpose of this article is to simplify the results of Yang [10 Yang , X. , Lu , C. ( 2000 ). Maximal properties of some subsemigroups in finite order-preserving transformation semigroups . Communications in Algebra 28 : 3125 – 3135 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]].     

Automorphism Groups of Schur Rings

In 1993, Muzychuk [23 Muzychuk , Mikhail E. ( 1993 ). The structure of rational Schur rings over cyclic groups . European Journal of Combinatorics 14 : 479 – 490 .[Crossref], [Web of Science ®] , [Google Scholar]] showed that the rational Schur rings over a cyclic group Z _n are in one-to-one correspondence with sublattices of the divisor lattice of n, or equivalently, with sublattices of the lattice of subgroups of Z _n . This can easily be extended to show that for any finite group G, sublattices of the lattice of characteristic subgroups of G give rise to rational Schur rings over G in a natural way. Our main result is that any finite group may be represented as the (algebraic) automorphism group of such a rational Schur ring over an abelian p-group, for any odd prime p. In contrast, over a cyclic group the automorphism group of any Schur ring is abelian. We also prove a converse to the well-known result of Muzychuk [24 Muzychuk , Mikhail E. ( 1994 ). On the structure of basic sets of Schur rings over cyclic groups . Journal of Algebra 169 : 655 – 678 .[Crossref], [Web of Science ®] , [Google Scholar]] that two Schur rings over a cyclic group are isomorphic if and only if they coincide; namely, we show that over a group which is not cyclic, there always exist distinct isomorphic Schur rings.     

Cluster Categories Coming from Cyclic Posets

Cyclic posets are generalizations of cyclically ordered sets. In this article, we show that any cyclic poset gives rise to a Frobenius category over any discrete valuation ring R. The stable category of a Frobenius category is always triangulated and has a cluster structure in many cases. The continuous cluster categories of [14 Igusa , K. , Todorov , G. ( 2013 ). Continuous Frobenius categories . Proceedings of the Abel Symposium 2011 : 115 – 143 .[Crossref] , [Google Scholar]], the infinity-gon of [12 Holm , T. , Jøot rgensen , P. ( 2012 ). On a cluster category of infinite Dynkin type, and the relation to triangulations of the infinity-gon . Math. Z. 270 : 277 – 295 .[Crossref], [Web of Science ®] , [Google Scholar]], and the m-cluster category of type A _∞ (m ≥ 3) [13 Holm , T. , Jøot rgensen , P. ( 2015 ). Cluster tilting vs. weak cluster tilting in Dynkin type A infinity . Forum Math. 27 : 1117 – 1137 .[Crossref], [Web of Science ®] , [Google Scholar]] are examples of this construction as well as some new examples such as the cluster category of ?². An extension of this construction and further examples are given in [16 Igusa, K., Todorov, G. Continuous Cluster Categories of Type D. arXiv:1309.7409 . [Google Scholar]].     

Rate of Convergence of a Restarted CG-DESCENT Method

In this article, we investigate the convergence rate of the CG-DESCENT method proposed by Hager and Zhang [1 W. W. Hager and H. Zhang ( 2005 ). A new conjugate gradient method with guaranteed descent and an effcient line search . SIAM Journal of Optimization 16 : 170 – 192 .[Crossref], [Web of Science ®] , [Google Scholar]]. Under reasonable conditions, we show that the CG-DESCENT method with the Wolfe line search will be n-step superlinear and even quadratic convergence if some restart technique is used. Some numerical results are also reported to verify the theoretical results.