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 Algebras and Representation Theory   [SJR: 0.868]   [H-I: 20]   [1 followers]  Follow         Hybrid journal (It can contain Open Access articles)    ISSN (Print) 1572-9079 - ISSN (Online) 1386-923X    Published by Springer-Verlag  [2352 journals]
• Applications of Normal Forms for Weighted Leavitt Path Algebras: Simple
Rings and Domains
• Authors: Roozbeh Hazrat; Raimund Preusser
Pages: 1061 - 1083
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)

• Gradings on Modules Over Lie Algebras of E Types
• Authors: Cristina Draper; Alberto Elduque; Mikhail Kochetov
Pages: 1085 - 1107
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)

• Varieties of Characters
• Authors: Sean Lawton; Adam S. Sikora
Pages: 1133 - 1141
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)

• A Characterization of Stem Algebras in Terms of Central Derivations
• Authors: F. Saeedi; S. Sheikh-Mohseni
Pages: 1143 - 1150
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: 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)

• Differential Modules over Quadratic Monomial Algebras
• Authors: Torkil Stai
Pages: 1239 - 1247
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)

• Locally Unmixed Modules and Linearly Equivalent Ideal Topologies
• Authors: Mona Bahadorian; Monireh Sedghi; Reza Naghipour
Pages: 1249 - 1257
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)

• Knuth’s Coherent Presentations of Plactic Monoids of Type A
• Authors: Nohra Hage; Philippe Malbos
Pages: 1259 - 1288
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)

• Irreducible Characters of p -group of Order ≤ p 5
• Authors: S. K. Prajapati; M. R. Darafsheh; M. Ghorbani
Pages: 1289 - 1303
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)

• Cosilting Modules
• Authors: Simion Breaz; Flaviu Pop
Pages: 1305 - 1321
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)

• Restricted Lie Algebras via Monadic Decomposition
• Authors: Alessandro Ardizzoni; Isar Goyvaerts; Claudia Menini
Abstract: We give a description of the category of restricted Lie algebras over a field $$\Bbbk$$ of prime characteristic by means of monadic decomposition of the functor that computes the $$\Bbbk$$ -vector space of primitive elements of a $$\Bbbk$$ -bialgebra.
PubDate: 2017-11-02
DOI: 10.1007/s10468-017-9734-8

• Levi Subgroup Actions on Schubert Varieties, Induced Decompositions of
their Coordinate Rings, and Sphericity Consequences
• Authors: Reuven Hodges; Venkatramani Lakshmibai
Abstract: Let L w be the Levi part of the stabilizer Q w in G L N (for left multiplication) of a Schubert variety X(w) in the Grassmannian G d,N . For the natural action of L w on $$\mathbb {C}[X(w)]$$ , the homogeneous coordinate ring of X(w) (for the Plücker embedding), we give a combinatorial description of the decomposition of $$\mathbb {C}[X(w)]$$ into irreducible L w -modules; in fact, our description holds more generally for the action of the Levi part L of any parabolic subgroup Q that is contained in Q w . This decomposition is then used to show that all smooth Schubert varieties, all determinantal Schubert varieties, and all Schubert varieties in G2,N are spherical L w -varieties.
PubDate: 2017-10-28
DOI: 10.1007/s10468-017-9744-6

• A Quantum Analog of Generalized Cluster Algebras
• Authors: Liqian Bai; Xueqing Chen; Ming Ding; Fan Xu
Abstract: We define a quantum analog of a class of generalized cluster algebras which can be viewed as a generalization of quantum cluster algebras defined in Berenstein and Zelevinsky (Adv. Math. 195(2), 405–455 2005). In the case of rank two, we extend some structural results from the classical theory of generalized cluster algebras obtained in Chekhov and Shapiro (Int. Math. Res. Notices 10, 2746–2772 2014) and Rupel (2013) to the quantum case.
PubDate: 2017-10-24
DOI: 10.1007/s10468-017-9743-7

• Irreducible Modules Over Witt Algebras W n $\mathcal {W}_{n}$ and Over
ð–˜ ð–‘ n + 1 ( ℂ ) $\mathfrak {sl}_{n+1}(\mathbb {C})$
• Authors: Haijun Tan; Kaiming Zhao
Abstract: In this paper, by using the “twisting technique” we obtain a class of new modules A b over the Witt algebras $$\mathcal {W}_{n}$$ from modules A over the Weyl algebras $$\mathcal {K}_{n}$$ (of Laurent polynomials) for any $$b\in \mathbb {C}$$ . We give necessary and sufficient conditions for A b to be irreducible, and determine necessary and sufficient conditions for two such irreducible $$\mathcal {W}_{n}$$ -modules to be isomorphic. Since $$\mathfrak {sl}_{n+1}(\mathbb {C})$$ is a subalgebra of $$\mathcal {W}_{n}$$ , all the above irreducible $$\mathcal {W}_{n}$$ -modules A b can be considered as $$\mathfrak {sl}_{n+1}(\mathbb {C})$$ -modules. For a class of such $$\mathfrak {sl}_{n+1}(\mathbb {C})$$ -modules, denoted by Ω1−a (λ 1, λ 2, ⋯ ,λ n ) where $$a\in \mathbb {C}, \lambda _{1},\lambda _{2},\cdots ,\lambda _{n} \in \mathbb {C}^{*}$$ , we determine necessary and sufficient conditions for these $$\mathfrak {sl}_{n+1}(\mathbb {C})$$ -modules to be irreducible. If the $$\mathfrak {sl}_{n+1}(\mathbb {C})$$ -module Ω1−a (λ 1, λ 2,⋯ ,λ n ) is reducible, we prove that it has a unique nontrivial submodule W 1−a (λ 1, λ 2,...λ n ) and the quotient module is the finite dimensional $$\mathfrak {sl}_{n+1}(\mathbb {C})$$ -module with highest weight mΛ n for some non-negative integer $$m\in \mathbb {Z}_{+}$$ . We also determine necessary and sufficient conditions for two $$\mathfrak {sl}_{n+1}(\mathbb {C})$$ -modules of the form Ω1−a (λ 1, λ 2,⋯ ,λ n ) or of the form W 1−a (λ 1, λ 2,...λ n ...
PubDate: 2017-10-19
DOI: 10.1007/s10468-017-9738-4

• The Injective Leavitt Complex
• Authors: Huanhuan Li
Abstract: For a finite quiver Q without sinks, we consider the corresponding finite dimensional algebra A with radical square zero. We construct an explicit compact generator for the homotopy category of acyclic complexes of injective A-modules. We call such a generator the injective Leavitt complex of Q. This terminology is justified by the following result: the differential graded endomorphism algebra of the injective Leavitt complex of Q is quasi-isomorphic to the Leavitt path algebra of Q. Here, the Leavitt path algebra is naturally $$\mathbb {Z}$$ -graded and viewed as a differential graded algebra with trivial differential.
PubDate: 2017-10-13
DOI: 10.1007/s10468-017-9741-9

• Dedualizing Complexes of Bicomodules and MGM Duality Over Coalgebras
• Authors: Leonid Positselski
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

• τ -Tilting Modules Over One-Point Extensions by a Projective Module
• Authors: Pamela Suarez
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

• Mutation of Torsion Pairs in Triangulated Categories and its Geometric
Realization
• Authors: Yu Zhou; Bin Zhu
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

• Irreducible Morphisms Between Modules over a Repetitive Algebras
• Authors: Hernán Giraldo
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

• Division Algebras with Left Algebraic Commutators
• Authors: M. Aaghabali; S. Akbari; M. H. Bien
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

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