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Publisher: Springer-Verlag   (Total: 2341 journals)

 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  [2341 journals]
• A New Approach to Representations of 3-Lie Algebras and Abelian Extensions
• Authors: Jiefeng Liu; Abdenacer Makhlouf; Yunhe Sheng
Abstract: Abstract In this paper, we introduce the notion of generalized representation of a 3-Lie algebra, by which we obtain a generalized semidirect product 3-Lie algebra. Moreover, we develop the corresponding cohomology theory. Various examples of generalized representations of 3-Lie algebras and computation of 2-cocycles of the new cohomology are provided. Also, we show that a split abelian extension of a 3-Lie algebra is isomorphic to a generalized semidirect product 3-Lie algebra. Furthermore, we describe general abelian extensions of 3-Lie algebras using Maurer-Cartan elements.
PubDate: 2017-04-17
DOI: 10.1007/s10468-017-9693-0

• Fusion Formulas and Fusion Procedure for the Yang-Baxter Equation
• Authors: L. Poulain d’Andecy
Abstract: Abstract We use the fusion formulas of the symmetric group and of the Hecke algebra to construct solutions of the Yang–Baxter equation on irreducible representations of $$\mathfrak {gl}_{N}$$ , $$\mathfrak {gl}_{N M}$$ , $$U_{q}(\mathfrak {gl}_{N})$$ and $$U_{q}(\mathfrak {gl}_{N M})$$ . The solutions are obtained via the fusion procedure for the Yang–Baxter equation, which is reviewed in a general setting. Distinguished invariant subspaces on which the fused solutions act are also studied in the general setting, and expressed, in general, with the help of a fusion function. Only then, the general construction is specialised to the four situations mentioned above. In each of these four cases, we show how the distinguished invariant subspaces are identified as irreducible representations, using the relevant fusion formula combined with the relevant Schur–Weyl duality.
PubDate: 2017-04-17
DOI: 10.1007/s10468-017-9692-1

• Degenerations of Submodules and Composition Series
• Authors: Nils M. Nornes; Steffen Oppermann
Abstract: Abstract Let M and N be modules over an artin algebra such that M degenerates to N. We show that any submodule of M degenerates to a submodule of N. This suggests that a composition series of M will in some sense degenerate to a composition series of N. We then study a subvariety of the module variety, consisting of those representations where all matrices are upper triangular. We show that these representations can be seen as representations of composition series, and that the orbit closures describe the above mentioned degeneration of composition series.
PubDate: 2017-04-04
DOI: 10.1007/s10468-017-9677-0

• Cosilting Modules
• Authors: Simion Breaz; Flaviu Pop
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: 2017-04-04
DOI: 10.1007/s10468-017-9688-x

• Authors: Alain Bruguières
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: 2017-04-03
DOI: 10.1007/s10468-017-9681-4

• Irreducible Characters of p -group of Order ≤ p 5
• Authors: S. K. Prajapati; M. R. Darafsheh; M. Ghorbani
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: 2017-03-31
DOI: 10.1007/s10468-017-9687-y

• Pro-Species of Algebras I: Basic Properties
• Authors: Julian Külshammer
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: 2017-03-31
DOI: 10.1007/s10468-017-9683-2

• Branching Rules for Finite-Dimensional U q ( ð–˜ ð–š ( 3 ) )
$\mathcal {U}_{q}(\mathfrak {su}(3))$ -Representations with Respect to a
Right Coideal Subalgebra
• Authors: Noud Aldenhoven; Erik Koelink; Pablo Román
Abstract: Abstract We consider the quantum symmetric pair $$(\mathcal {U}_{q}(\mathfrak {su}(3)), \mathcal {B})$$ where $$\mathcal {B}$$ is a right coideal subalgebra. We prove that all finite-dimensional irreducible representations of $$\mathcal {B}$$ are weight representations and are characterised by their highest weight and dimension. We show that the restriction of a finite-dimensional irreducible representation of $$\mathcal {U}_{q}(\mathfrak {su}(3))$$ to $$\mathcal {B}$$ decomposes multiplicity free into irreducible representations of $$\mathcal {B}$$ . Furthermore we give explicit expressions for the highest weight vectors in this decomposition in terms of dual q-Krawtchouk polynomials.
PubDate: 2017-03-25
DOI: 10.1007/s10468-017-9678-z

• Differential Modules over Quadratic Monomial Algebras
• Authors: Torkil Stai
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: 2017-03-24
DOI: 10.1007/s10468-017-9684-1

• Locally Unmixed Modules and Linearly Equivalent Ideal Topologies
• Authors: Mona Bahadorian; Monireh Sedghi; Reza Naghipour
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: 2017-03-24
DOI: 10.1007/s10468-017-9685-0

• Varieties of Characters
• Authors: Sean Lawton; Adam S. Sikora
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: 2017-03-18
DOI: 10.1007/s10468-017-9679-y

• K 0 of Semiartinian Von Neumann Regular Rings. Direct Finiteness Versus
Unit-Regularity
• Authors: Giuseppe Baccella; Leonardo Spinosa
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: 2017-03-13
DOI: 10.1007/s10468-017-9682-3

• A Characterization of Stem Algebras in Terms of Central Derivations
• Authors: F. Saeedi; S. Sheikh-Mohseni
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: 2017-03-10
DOI: 10.1007/s10468-017-9680-5

• Gradings on Modules Over Lie Algebras of E Types
• Authors: Cristina Draper; Alberto Elduque; Mikhail Kochetov
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: 2017-03-06
DOI: 10.1007/s10468-017-9675-2

• Refinements of the Orthogonality Relations for Blocks
• Authors: Benjamin Sambale
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: 2017-03-06
DOI: 10.1007/s10468-017-9676-1

• Locally Quadratic Modules and Minuscule Representations
Abstract: We give a new, geometric proof of a theorem by Timmesfeld showing that for simple Chevalley groups, abstract modules where all roots act quadratically are direct sums of minuscule representations. Our proof is uniform, treats finite and infinite fields on an equal footing, and includes Lie rings.
PubDate: 2017-03-06
DOI: 10.1007/s10468-017-9671-6

• Applications of Normal Forms for Weighted Leavitt Path Algebras: Simple
Rings and Domains
• Authors: Roozbeh Hazrat; Raimund Preusser
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: 2017-02-25
DOI: 10.1007/s10468-017-9674-3

• The Representation Type of Determinantal Varieties
• Authors: Jan O. Kleppe; Rosa M. Miró-Roig
Abstract: Abstract This work is entirely devoted to construct huge families of indecomposable arithmetically Cohen-Macaulay (resp. Ulrich) sheaves $$\mathcal {E}$$ of arbitrary high rank on a general standard (resp. linear) determinantal scheme $$X\subset \mathbb {P}^{n}$$ of codimension c ≥ 1, n − c ≥ 1 and defined by the maximal minors of a t × (t + c−1) homogeneous matrix $$\mathcal {A}$$ . The sheaves $$\mathcal {E}$$ are constructed as iterated extensions of sheaves of lower rank. As applications: (1) we prove that any general standard determinantal scheme $$X\subset \mathbb {P}^{n}$$ is of wild representation type provided the degrees of the entries of the matrix $$\mathcal {A}$$ satisfy some weak numerical assumptions; and (2) we determine values of t, n and n − c for which a linear standard determinantal scheme $$X\subset \mathbb {P}^{n}$$ is of wild representation type with respect to the much more restrictive category of its indecomposable Ulrich sheaves, i.e. X is of Ulrich wild representation type.
PubDate: 2017-02-25
DOI: 10.1007/s10468-017-9673-4

• On Frobenius (Completed) Orbit Categories
• Authors: Alfredo Nájera Chávez
Abstract: Abstract Let ð“” be a Frobenius category, $${\mathcal P}$$ its subcategory of projective objects and F : ð“” → ð“” an exact automorphism. We prove that there is a fully faithful functor from the orbit category ð“”/F into $$\operatorname {gpr}({\mathcal P}/F)$$ , the category of finitely-generated Gorenstein-projective modules over $${\mathcal P}/F$$ . We give sufficient conditions to ensure that the essential image of ð“”/F is an extension-closed subcategory of $$\operatorname {gpr}({\mathcal P}/F)$$ . If ð“” is in addition Krull-Schmidt, we give sufficient conditions to ensure that the completed orbit category $${\mathcal E} \ \widehat {\!\! /} F$$ is a Krull-Schmidt Frobenius category. Finally, we apply our results on completed orbit categories to the context of Nakajima categories associated to Dynkin quivers and sketch applications to cluster algebras.
PubDate: 2017-02-23
DOI: 10.1007/s10468-017-9672-5

• Whittaker Modules for the Insertion-Elimination Lie Algebra
• Authors: Matthew Ondrus; Emilie Wiesner
Abstract: Abstract This paper addresses the representation theory of the insertion-elimination Lie algebra, a Lie algebra that can be naturally realized in terms of tree-inserting and tree-eliminating operations on rooted trees. The insertion-elimination algebra admits a triangular decomposition in the sense of Moody and Pianzola, and thus it is natural to define Whittaker modules corresponding to a given algebra homomorphism. Among other results, we show that the standard Whittaker modules are simple under certain constraints on the corresponding algebra homomorphism.
PubDate: 2017-02-22
DOI: 10.1007/s10468-016-9665-9

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