Authors:Roozbeh Hazrat; Raimund Preusser Pages: 1061 - 1083 Abstract: Abstract Weighted Leavitt path algebras (wLpas) are a generalisation of Leavitt path algebras (with graphs of weight 1) and cover the algebras L K (n, n + k) constructed by Leavitt. Using Bergman’s diamond lemma, we give normal forms for elements of a weighted Leavitt path algebra. This allows us to produce a basis for a wLpa. Using the normal form we classify the wLpas which are domains, simple and graded simple rings. For a large class of weighted Leavitt path algebras we establish a local valuation and as a consequence we prove that these algebras are prime, semiprimitive and nonsingular but contrary to Leavitt path algebras, they are not graded von Neumann regular. PubDate: 2017-10-01 DOI: 10.1007/s10468-017-9674-3 Issue No:Vol. 20, No. 5 (2017)

Authors:Cristina Draper; Alberto Elduque; Mikhail Kochetov Pages: 1085 - 1107 Abstract: Abstract For any grading by an abelian group G on the exceptional simple Lie algebra \(\mathcal {L}\) of type E 6 or E 7 over an algebraically closed field of characteristic zero, we compute the graded Brauer invariants of simple finite-dimensional modules, thus completing the computation of these invariants for simple finite-dimensional Lie algebras. This yields the classification of finite-dimensional G-graded simple \(\mathcal {L}\) -modules, as well as necessary and sufficient conditions for a finite-dimensional \(\mathcal {L}\) -module to admit a G-grading compatible with the given G-grading on \(\mathcal {L}\) . PubDate: 2017-10-01 DOI: 10.1007/s10468-017-9675-2 Issue No:Vol. 20, No. 5 (2017)

Authors:Benjamin Sambale Pages: 1109 - 1131 Abstract: Abstract For a block B of a finite group G there are well-known orthogonality relations for the generalized decomposition numbers. We refine these relations by expressing the generalized decomposition numbers with respect to an integral basis of a certain cyclotomic field. After that, we use the refinements in order to give upper bounds for the number of irreducible characters (of height 0) in B. In this way we generalize results from [Héthelyi-Külshammer-Sambale, 2014]. These ideas are applied to blocks with abelian defect groups of rank 2. Finally, we address a recent conjecture by Navarro. PubDate: 2017-10-01 DOI: 10.1007/s10468-017-9676-1 Issue No:Vol. 20, No. 5 (2017)

Authors:Sean Lawton; Adam S. Sikora Pages: 1133 - 1141 Abstract: Abstract Let G be a connected reductive affine algebraic group. In this short note we define the variety of G-characters of a finitely generated group Γ and show that the quotient of the G-character variety of Γ by the action of the trace preserving outer automorphisms of G normalizes the variety of G-characters when Γ is a free group, free abelian group, or a surface group. PubDate: 2017-10-01 DOI: 10.1007/s10468-017-9679-y Issue No:Vol. 20, No. 5 (2017)

Authors:F. Saeedi; S. Sheikh-Mohseni Pages: 1143 - 1150 Abstract: Abstract Let L be a Lie algebra, and Der z (L) denote the set of all central derivations of L, that is, the set of all derivations of L mapping L into the center. In this paper, by using the notion of isoclinism, we study the center of Der z (L) for nilpotent Lie algebras with nilindex 2. We also give a characterization of stem Lie algebras by their central derivations. In fact we show that for non-abelian nilpotent Lie algebras of finite dimension and any nilpotent Lie algebra with nilindex 2 (not finite dimensional in general), Der z (L) is abelian if and only if L is a stem Lie algebra. PubDate: 2017-10-01 DOI: 10.1007/s10468-017-9680-5 Issue No:Vol. 20, No. 5 (2017)

Authors:Alain Bruguières Pages: 1151 - 1188 Abstract: Abstract We introduce Hopf polyads in order to unify Hopf monads and group actions on monoidal categories. A polyad is a lax functor from a small category (its source) to the bicategory of categories, and a Hopf polyad is a comonoidal polyad whose fusion operators are invertible. The main result states that the normalization of a Hopf polyad is a strong (co)monoidal action-type polyad (or strong monoidal pseudofunctor). The normalization of a polyad is a new polyad having simpler structure but the same category of modules. We show that, under certain assumptions, a Hopf polyad can be ‘wrapped up’ into a Hopf monad. This generalizes the fact that finite group actions on tensor categories can be seen as Hopf monads. Hopf categories in the sense of Batista, Caenepeel and Vercruysse can be viewed as Hopf polyads in a braided setting via the notion of Hopf polyalgebras. As a special case of the main theorem, we generalize a description of the center of graded fusion category due to Turaev and Virelizier to tensor categories: if \(\mathcal {C}\) is a G-graded (locally bounded) tensor category, then G acts on the relative center of \(\mathcal {C}\) with respect to the degree one part \(\mathcal {C}_{1}\) , and the equivariantization of this action is the center of \(\mathcal {C}\) . PubDate: 2017-10-01 DOI: 10.1007/s10468-017-9681-4 Issue No:Vol. 20, No. 5 (2017)

Authors:Giuseppe Baccella; Leonardo Spinosa Pages: 1189 - 1213 Abstract: Abstract If R is a regular and semiartinian ring, it is proved that the following conditions are equivalent: (1) R is unit-regular, (2) every factor ring of R is directly finite, (3) the abelian group K O(R) is free and admits a basis which is in a canonical one to one correspondence with a set of representatives of simple right R-modules. For the class of semiartinian and unit-regular rings the canonical partial order of K O(R) is investigated. Starting from any partially ordered set I, a special dimension group G(I) is built and a large class of semiartinian and unit-regular rings is shown to have the corresponding K O(R) order isomorphic to G(P r i m R ), where P r i m R is the primitive spectrum of R. Conversely, if I is an artinian partially ordered set having a finite cofinal subset, it is proved that the dimension group G(I) is realizable as K O(R) for a suitable semiartinian and unit-regular ring R. PubDate: 2017-10-01 DOI: 10.1007/s10468-017-9682-3 Issue No:Vol. 20, No. 5 (2017)

Authors:Julian Külshammer Pages: 1215 - 1238 Abstract: Abstract In this paper, we generalise part of the theory of hereditary algebras to the context of pro-species of algebras. Here, a pro-species is a generalisation of Gabriel’s concept of species gluing algebras via projective bimodules along a quiver to obtain a new algebra. This provides a categorical perspective on a recent paper by Geiß et al. (2016). In particular, we construct a corresponding preprojective algebra, and establish a theory of a separated pro-species yielding a stable equivalence between certain functorially finite subcategories. PubDate: 2017-10-01 DOI: 10.1007/s10468-017-9683-2 Issue No:Vol. 20, No. 5 (2017)

Authors:Torkil Stai Pages: 1239 - 1247 Abstract: Abstract We compare the so-called clock condition to the gradability of certain differential modules over quadratic monomial algebras. These considerations show that a stably hereditary or gentle one-cycle algebra is piecewise hereditary if and only if the orbit category of its bounded derived category with respect to a positive power of the shift functor is triangulated. PubDate: 2017-10-01 DOI: 10.1007/s10468-017-9684-1 Issue No:Vol. 20, No. 5 (2017)

Authors:Mona Bahadorian; Monireh Sedghi; Reza Naghipour Pages: 1249 - 1257 Abstract: Abstract Let R be a commutative Noetherian ring, and let N be a non-zero finitely generated R-module. The purpose of this paper is to show that N is locally unmixed if and only if, for any N-proper ideal I of R generated by ht N I elements, the topology defined by (I N)(n), n ≥ 0, is linearly equivalent to the I-adic topology. PubDate: 2017-10-01 DOI: 10.1007/s10468-017-9685-0 Issue No:Vol. 20, No. 5 (2017)

Authors:Nohra Hage; Philippe Malbos Pages: 1259 - 1288 Abstract: Abstract We construct finite coherent presentations of plactic monoids of type A. Such coherent presentations express a system of generators and relations for the monoid extended in a coherent way to give a family of generators of the relations amongst the relations. Such extended presentations are used for representations of monoids, in particular, it is a way to describe actions of monoids on categories. Moreover, a coherent presentation provides the first step in the computation of a categorical cofibrant replacement of a monoid. Our construction is based on a rewriting method introduced by Squier that computes a coherent presentation from a convergent one. We compute a finite coherent presentation of a plactic monoid from its column presentation and we reduce it to a Tietze equivalent one having Knuth’s generators. PubDate: 2017-10-01 DOI: 10.1007/s10468-017-9686-z Issue No:Vol. 20, No. 5 (2017)

Authors:S. K. Prajapati; M. R. Darafsheh; M. Ghorbani Pages: 1289 - 1303 Abstract: Abstract P. Hall introduced the concept of isoclinism of groups to classify p-groups. It is well-known that two isoclinic nilpotent groups have the same nilpotency class. In this paper using the classification of James of p-groups of order at most p 5 via their isoclinism classes, the degrees of irreducible characters with their frequencies are found. To do this we use the concept of generalized Camina pairs. We also investigate that whether a nonlinear irreducible character can be obtained as a product of two other nonlinear irreducible characters of same degree. PubDate: 2017-10-01 DOI: 10.1007/s10468-017-9687-y Issue No:Vol. 20, No. 5 (2017)

Authors:Simion Breaz; Flaviu Pop Pages: 1305 - 1321 Abstract: Abstract We study the class of modules, called cosilting modules, which are defined as the categorical duals of silting modules. Several characterizations of these modules and connections with silting modules are presented. We prove that Bazzoni’s theorem about the pure-injectivity of cotilting modules is also valid for cosilting modules. PubDate: 2017-10-01 DOI: 10.1007/s10468-017-9688-x Issue No:Vol. 20, No. 5 (2017)

Authors:Leonid Positselski Abstract: Abstract We present the definition of a dedualizing complex of bicomodules over a pair of cocoherent coassociative coalgebras \(\mathcal {C}\) and \(\mathcal {D}\) . Given such a complex \(\mathcal {B}^{\bullet }\) , we construct an equivalence between the (bounded or unbounded) conventional, as well as absolute, derived categories of the abelian categories of left comodules over \(\mathcal {C}\) and left contramodules over \(\mathcal {D}\) . Furthermore, we spell out the definition of a dedualizing complex of bisemimodules over a pair of semialgebras, and construct the related equivalence between the conventional or absolute derived categories of the abelian categories of semimodules and semicontramodules. Artinian, co-Noetherian, and cocoherent coalgebras are discussed as a preliminary material. PubDate: 2017-10-12 DOI: 10.1007/s10468-017-9736-6

Authors:Pamela Suarez Abstract: Abstract Let A be the one point extension of an algebra B by a projective B-module. We prove that the extension of a given support τ-tilting B-module is a support τ-tilting A-module; and, conversely, the restriction of a given support τ-tilting A-module is a support τ-tilting B-module. Moreover, we prove that there exists a full embedding of quivers between the corresponding poset of support τ-tilting modules. PubDate: 2017-10-07 DOI: 10.1007/s10468-017-9737-5

Authors:Yu Zhou; Bin Zhu Abstract: Abstract We introduce and study mutation of torsion pairs, as a generalization of mutation of cluster tilting objects, rigid objects and maximal rigid objects. It is proved that any mutation of a torsion pair is again a torsion pair. A geometric realization of mutation of torsion pairs in the cluster category of type A n or A ∞ is given via rotation of Ptolemy diagrams. PubDate: 2017-10-06 DOI: 10.1007/s10468-017-9740-x

Authors:Hernán Giraldo Abstract: Abstract We describe the irreducible morphisms in the category of modules over a repetitive algebra. We find three special canonical forms: The first canonical form happens when all the component morphisms are split monomorphisms, the second when all the component morphisms are split epimorphisms and the third when there is exactly one irreducible component map. Also, we obtain the same result for the irreducible homomorphisms in the stable category of modules over a repetitive algebra. PubDate: 2017-10-05 DOI: 10.1007/s10468-017-9733-9

Authors:M. Aaghabali; S. Akbari; M. H. Bien Abstract: Abstract Let D be a division algebra with center F and K a (not necessarily central) subfield of D. An element a ∈ D is called left algebraic (resp. right algebraic) over K, if there exists a non-zero left polynomial a 0 + a 1 x + ⋯ + a n x n (resp. right polynomial a 0 + x a 1 + ⋯ + x n a n ) over K such that a 0 + a 1 a + ⋯ + a n a n = 0 (resp. a 0 + a a 1 + ⋯ + a n a n ). Bell et al. proved that every division algebra whose elements are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. In this paper we generalize this result and prove that every division algebra whose all multiplicative commutators are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite provided that the center of division algebra is infinite. Also, we show that every division algebra whose multiplicative group of commutators is left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. Among other results we present similar result regarding additive commutators under certain conditions. PubDate: 2017-10-03 DOI: 10.1007/s10468-017-9739-3

Authors:Amnon Yekutieli Abstract: Abstract We continue investigating the interaction between flatness and \({\frak{a}} \) -adic completion for infinitely generated A-modules. Here A is a commutative ring and \({\frak{a}} \) is a finitely generated ideal in it. We introduce the concept of \({\frak{a}} \) -adic flatness, which is weaker than flatness. We prove that \({\frak{a}} \) -adic flatness is preserved under completion when the ideal \({\frak{a}} \) is weakly proregular. We also prove that when A is noetherian, \({\frak{a}} \) -adic flatness coincides with flatness (for complete modules). An example is worked out of a non-noetherian ring A, with a weakly proregular ideal \({\frak{a}} \) , for which the completion \(\widehat {A}\) is not flat. We also study \({\frak{a}} \) -adic systems, and prove that if the ideal \({\frak{a}} \) is finitely generated, then the limit of every \({\frak{a}} \) -adic system is a complete module. PubDate: 2017-09-23 DOI: 10.1007/s10468-017-9735-7

Authors:Sota Asai Abstract: Abstract We deal with the finite-dimensional mesh algebras given by stable translation quivers. These algebras are self-injective, and thus the stable module categories have a structure of triangulated categories. Our main result determines the Grothendieck groups of these stable module categories. As an application, we give a complete classification of the mesh algebras up to stable equivalences. PubDate: 2017-09-07 DOI: 10.1007/s10468-017-9732-x