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Jun 26, 2018
06/18
by
Benjamin Steinberg
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The representation theory of the symmetric group has been intensively studied for over 100 years and is one of the gems of modern mathematics. The full transformation monoid $\mathfrak T_n$ (the monoid of all self-maps of an $n$-element set) is the monoid analogue of the symmetric group. The investigation of its representation theory was begun by Hewitt and Zuckerman in 1957. Its character table was computed by Putcha in 1996 and its representation type was determined in a series of papers by...
Topics: Mathematics, Representation Theory, Rings and Algebras, Group Theory
Source: http://arxiv.org/abs/1502.00959
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Jun 29, 2018
06/18
by
Ruari Walker
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This paper is investigative work into the properties of a family of graded algebras recently defined by Varagnolo and Vasserot, which we call VV algebras. We compare categories of modules over KLR algebras with categories of modules over VV algebras, establishing various Morita equivalences. Using these Morita equivalences we are able to prove several properties of certain classes of VV algebras such as (graded) affine cellularity and affine quasi-heredity.
Topics: Representation Theory, Rings and Algebras, Mathematics
Source: http://arxiv.org/abs/1603.00796
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59
Jun 29, 2018
06/18
by
Andrew Dolphin; Anne Quéguiner-Mathieu
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In this paper we study symplectic involutions and quadratic pairs that become hyperbolic over the function field of a conic. In particular, we classify them in degree 4 and deduce results on 5 dimensional minimal quadratic forms, thus extending to arbitrary fields some results of [24], which were only known in characteristic different from 2.
Topics: Rings and Algebras, Mathematics
Source: http://arxiv.org/abs/1604.04733
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Jun 26, 2018
06/18
by
Sanghoon Baek
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We study the semi-decomposable invariants of a split semisimple group and their extension to a split reductive group by using the torsion in the codimension $2$ Chow groups of a product of Severi-Brauer varieties. In particular, for any $n\geq 2$ we completely determine the degree $3$ invariants of a split semisimple group, the quotient of $(\operatorname{\mathbf{SL}}_{2})^{n}$ by its maximal central subgroup, as well as of the corresponding split reductive group. We also provide an example...
Topics: Algebraic Geometry, Mathematics, Rings and Algebras
Source: http://arxiv.org/abs/1502.03023
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Jun 29, 2018
06/18
by
Lauren Grimley
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We formulate the Gerstenhaber algebra structure of Hochschild cohomology of finite group extensions of some quantum complete intersections. When the group is trivial, this work characterizes the graded Lie brackets on Hochschild cohomology of these quantum complete intersections, previously only known for a few cases. As an example, we compute the algebra structure for two generator quantum complete intersections extended by select groups.
Topics: Rings and Algebras, Representation Theory, Quantum Algebra, Mathematics
Source: http://arxiv.org/abs/1606.01727
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Jun 26, 2018
06/18
by
Rod Gow
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K be a field and let m and n be positive integers, where m does not exceed n. We say that a non-zero subspace of m x n matrices over K is a constant rank r subspace if each non-zero element of the subspace has rank r, where r is a positive integer that does not exceed m. We show in this paper that if K is a finite field containing at least r+1 elements, any constant rank r subspace of m x n matrices over K has dimension at most n.
Topics: Rings and Algebras, Mathematics
Source: http://arxiv.org/abs/1501.02721
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Jun 28, 2018
06/18
by
A. Tsurkov
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In this paper we consider the very wide class of varieties of representations of Lie algebras over the field k, which has characteristic 0. We study the relation between the geometric equivalence and automorphic equivalence of the representations of these varieties. We calculate the group, which measures the difference between the geometric equivalence and automorphic equivalence of representations of theses varieties. In Section 5, we present one example of the subvariety of the variety of all...
Topics: Mathematics, Rings and Algebras
Source: http://arxiv.org/abs/1508.03034
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Jun 28, 2018
06/18
by
Attila Nagy
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In this paper we show that if $I$ is an ideal of a commutative semigroup $C$ such that the separator $SepI$ of $I$ is not empty then the factor semigroup $S=C/P_I$ ($P_I$ is the principal congruence on $C$ defined by $I$) satisfies Condition $(*)$: $S$ is a commutative monoid with a zero; The annihilator $A(s)$ of every non identity element $s$ of $S$ contains a non zero element of $S$; $A(s)=A(t)$ implies $s=t$ for every $s, t\in S$. Conversely, if $\alpha$ is a congruence on a commutative...
Topics: Group Theory, Number Theory, Mathematics, Rings and Algebras
Source: http://arxiv.org/abs/1508.07430
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Jun 28, 2018
06/18
by
Juan Cuadra; Pavel Etingof; Chelsea Walton
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We prove that any action of a finite dimensional Hopf algebra H on a Weyl algebra A over an algebraically closed field of characteristic zero factors through a group action. In other words, Weyl algebras do not admit genuine finite quantum symmetries. This improves a previous result by the authors, where the statement was established for semisimple H. The proof relies on a refinement of the method previously used: namely, considering reductions of the action of H on A modulo prime powers rather...
Topics: Mathematics, Rings and Algebras, Quantum Algebra
Source: http://arxiv.org/abs/1509.01165
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Jun 30, 2018
06/18
by
Benjamin Steinberg
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A well-known theorem of Burnside says that if $\rho$ is a faithful representation of a finite group $G$ over a field of characteristic $0$, then every irreducible representation of $G$ appears as a constituent of a tensor power of $\rho$. In 1962, R. Steinberg gave a module theoretic proof that simultaneously removed the constraint on the characteristic, and allowed the group to be replaced by a monoid. Brauer subsequently simplified Burnside's proof and, moreover, showed that if the character...
Topics: Mathematics, Rings and Algebras, Representation Theory, Group Theory
Source: http://arxiv.org/abs/1409.7632
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Jun 30, 2018
06/18
by
Ralf Schiffler; Khrystyna Serhiyenko
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We propose a new approach to study the relation between the module categories of a tilted algebra $C$ and the corresponding cluster-tilted algebra $B=C\ltimes E$. This new approach consists of using the induction functor $-\otimes_C B$ as well as the coinduction functor $D(B\otimes_C D-)$. We show that $DE$ is a partial tilting and a $\tau$-rigid $C$-module and that the induced module $DE\otimes_C B$ is a partial tilting and a $\tau$-rigid $B$-module. Furthermore, if $C=\text{End}_A T$ for a...
Topics: Mathematics, Rings and Algebras, Representation Theory
Source: http://arxiv.org/abs/1410.1732
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Jun 27, 2018
06/18
by
Jiaqun Wei
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We study singularity categories through Gorenstein objects in triangulated categories and silting theory. Let ${\omega}$ be a semi-selforthogonal (or presilting) subcategory of a triangulated category $\mathcal{T}$. We introduce the notion of $\omega$-Gorenstein objects, which is far extended version of Gorenstein projective modules and Gorenstein injective modules in triangulated categories. We prove that the stable category $\underline{\mathcal{G}_{\omega}}$, where $\mathcal{G}_{\omega}$ is...
Topics: Rings and Algebras, Category Theory, Mathematics, Algebraic Geometry, Representation Theory,...
Source: http://arxiv.org/abs/1504.06738
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Jun 27, 2018
06/18
by
Van C. Nguyen; Linhong Wang; Xingting Wang
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For finite-dimensional Hopf algebras, their classification in characteristic $0$ (e.g. over $\mathbb{C}$) has been investigated for decades with many fruitful results, but their structures in positive characteristic have remained elusive. In this paper, working over an algebraically closed field $\mathbf{k}$ of prime characteristic $p$, we introduce the concept, called Primitive Deformation, to provide a structured technique to classify certain finite-dimensional connected Hopf algebras which...
Topics: Mathematics, Rings and Algebras
Source: http://arxiv.org/abs/1505.02454
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Jun 27, 2018
06/18
by
Rafail Alizade; Engin Buyukasik
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In this paper, poor abelian groups are characterized. It is proved that an abelian group is poor if and only if its torsion part contains a direct summand isomorphic to $\oplus_{p \in P} \Z_p$, where $P$ is the set of prime integers. We also prove that pi-poor abelian groups exist. Namely, it is proved that the direct sum of $U^{(\mathbb{N})}$, where $U$ ranges over all nonisomorphic uniform abelian groups, is pi-poor. Moreover, for a pi-poor abelian group $M$, it is shown that $M$ can not be...
Topics: Mathematics, Rings and Algebras
Source: http://arxiv.org/abs/1505.03300
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7.0
Jun 30, 2018
06/18
by
Gurmeet K. Bakshi; Gurleen Kaur
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In this paper, a construction of Shoda pairs using character triples is given for a large class of monomial groups including abelian-by-supersolvable and subnormally monomial groups. The computation of primitive central idempotents and the structure of simple components of the rational group algebra for groups in this class are also discussed. The theory is illustrated with examples.
Topics: Group Theory, Rings and Algebras, Mathematics
Source: http://arxiv.org/abs/1702.00955
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Jun 30, 2018
06/18
by
Tiwei Zhao
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In this paper, we define a class of relative derived functors in terms of left or right weak flat resolutions to compute the weak flat dimension of modules. Moreover, we investigate two classes of modules larger than that of weak injective and weak flat modules, study the existence of covers and preenvelopes, and give some applications.
Topics: Rings and Algebras, Category Theory, Mathematics
Source: http://arxiv.org/abs/1704.02456
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Jun 30, 2018
06/18
by
Jan Stovicek
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Given a locally coherent Grothendieck category G, we prove that the homotopy category of complexes of injective objects (also known as the coderived category of G) is compactly generated triangulated. Moreover, the full subcategory of compact objects is none other than D^b(fp G). If G admits a generating set of finitely presentable objects of finite projective dimension, then also the derived category of G is compactly generated and Krause's recollement exists. Our main tools are (a) model...
Topics: Category Theory, Mathematics, Rings and Algebras, Representation Theory, Algebraic Geometry
Source: http://arxiv.org/abs/1412.1615
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6.0
Jun 30, 2018
06/18
by
Victoria Gould; Miklos Hartmann; Nik Ruskuc
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A monoid $S$ is said to be right coherent if every finitely generated subact of every finitely presented right $S$-act is finitely presented. Left coherency is defined dually and $S$ is coherent if it is both right and left coherent. These notions are analogous to those for a ring $R$ (where, of course, $S$-acts are replaced by $R$-modules). Choo, Lam and Luft have shown that free rings are coherent. In this note we prove that, correspondingly, any free monoid is coherent, thus answering a...
Topics: Mathematics, Rings and Algebras
Source: http://arxiv.org/abs/1412.7340
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Jun 25, 2018
06/18
by
Mahmood Alizadeh
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Recently, the $k$-normal element over finite fields is defined and characterized by Huczynska et al.. In this paper, the characterization of $k$-normal elements, by using to give a generalization of Schwartz's theorem, which allows us to check if an element is a normal element, is obtained. In what follows, in respect of the problem of existence of a primitive 1-normal element in $\mathbb{F}_{q^n}$ over $\mathbb{F}_{q}$, for all $q$ and $n$, had been stated by Huczynska et al., it is shown...
Topics: Commutative Algebra, Mathematics, Rings and Algebras, Mathematics
Source: http://arxiv.org/abs/1501.00397
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Jun 27, 2018
06/18
by
Diana Savin
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In this paper we determine some properties of Fibonacci octonions. Also, we introduce the generalized Fibonacci-Lucas octonions and we investigate some properties of these elements.
Topics: Combinatorics, Mathematics, Rings and Algebras
Source: http://arxiv.org/abs/1505.01770
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24
Jun 27, 2018
06/18
by
Clément de Seguins Pazzis
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Let $U$ and $V$ be finite-dimensional vector spaces over an arbitrary field, and $\mathcal{S}$ be a subset of the space $\mathcal{L}(U,V)$ of all linear maps from $U$ to $V$. A map $F : \mathcal{S} \rightarrow V$ is called range-compatible when it satisfies $F(s) \in \mathrm{im}(s)$ for all $s \in \mathcal{S}$; it is called quasi-range-compatible when the condition is only assumed to apply to the operators whose range does not include a fixed $1$-dimensional linear subspace of $V$. Among the...
Topics: Mathematics, Rings and Algebras
Source: http://arxiv.org/abs/1505.02315
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Jun 28, 2018
06/18
by
Friedrich Martin Schneider; Jens Zumbrägel
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A Hausdorff topological semiring is called simple if every non-zero continuous homomorphism into another Hausdorff topological semiring is injective. Classical work by Anzai and Kaplansky implies that any simple compact ring is finite. We generalize this result by proving that every simple compact semiring is finite, i.e., every infinite compact semiring admits a proper non-trivial quotient.
Topics: Mathematics, Rings and Algebras, General Topology
Source: http://arxiv.org/abs/1509.01133
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Jun 29, 2018
06/18
by
Genqiang Liu; Rencai Lu; Kaiming Zhao
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For an irreducible module $P$ over the Weyl algebra $\mathcal{K}_n^+$ (resp. $\mathcal{K}_n$) and an irreducible module $M$ over the general liner Lie algebra $\mathfrak{gl}_n$, using Shen's monomorphism, we make $P\otimes M$ into a module over the Witt algebra $W_n^+$ (resp. over $W_n$). We obtain the necessary and sufficient conditions for $P\otimes M$ to be an irreducible module over $W_n^+$ (resp. $W_n$), and determine all submodules of $P\otimes M$ when it is reducible. Thus we have...
Topics: Rings and Algebras, Representation Theory, Quantum Algebra, Mathematics
Source: http://arxiv.org/abs/1612.00315
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Jun 30, 2018
06/18
by
Huanyin Chen
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An element of a ring is unique clean if it can be uniquely written as the sum of an idempotent and a unit. A ring $R$ is uniquely $\pi$-clean if some power of every element in $R$ is uniquely clean. In this article, we prove that a ring $R$ is uniquely $\pi$-clean if and only if for any $a\in R$, there exists an $m\in {\Bbb N}$ and a central idempotent $e\in R$ such that $a^m-e\in J(R)$, if and only if $R$ is abelian; every idempotent lifts modulo $J(R)$; and $R/P$ is torsion for all prime...
Topics: Mathematics, Rings and Algebras
Source: http://arxiv.org/abs/1406.7472
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49
Jun 30, 2018
06/18
by
Birge Huisgen-Zimmermann
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Given a finite dimensional representation $M$ of a finite dimensional algebra, two hierarchies of degenerations of $M$ are analyzed in the context of their natural orders: the poset of those degenerations of $M$ which share the top $M/JM$ with $M$ - here $J$ denotes the radical of the algebra - and the sub-poset of those which share the full radical layering $\bigl(J^lM/J^{l+1}M\bigr)_{l \ge 0}$ with $M$. In particular, the article addresses existence of proper top-stable or layer-stable...
Topics: Mathematics, Rings and Algebras, Representation Theory
Source: http://arxiv.org/abs/1407.2665
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Jun 30, 2018
06/18
by
Raf Bocklandt; Federica Galluzzi; Francesco Vaccarino
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Let $k$ be an algebraically closed field of characteristic zero and let $A$ be a finitely generated $k-$algebra. The Nori - Hilbert scheme of $A$, parameterizes left ideals of codimension $n$ in $A,$ and it is well known to be smooth when $A$ is formally smooth. In this paper we will study the Nori - Hilbert scheme for $2-$Calabi Yau algebras. The main examples of these are surface group algebras and preprojective algebras. For the former we show that the Nori-Hilbert scheme is smooth for $n=1$...
Topics: Mathematics, Rings and Algebras, Representation Theory, Algebraic Geometry
Source: http://arxiv.org/abs/1410.1442
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Jun 29, 2018
06/18
by
Florian Eisele; Michael Geline; Radha Kessar; Markus Linckelmann
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We identify a class of symmetric algebras over a complete discrete valuation ring $\mathcal O$ of characteristic zero to which the characterisation of Kn\"orr lattices in terms of stable endomorphism rings in the case of finite group algebras, can be extended. This class includes finite group algebras, their blocks and source algebras and Hopf orders. We also show that certain arithmetic properties of finite group representations extend to this class of algebras. Our results are based on...
Topics: Representation Theory, Rings and Algebras, Mathematics
Source: http://arxiv.org/abs/1608.06497
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6.0
Jun 29, 2018
06/18
by
Ann Kiefer
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We generalize an algorithm established in earlier work to compute finitely many generators for a subgroup of finite index of a group acting discontinuously on hyperbolic space of dimension $2$ and $3$, to hyperbolic space of higher dimensions using Clifford algebras. We hence get an algorithm which gives a finite set of generators up to finite index of a discrete subgroup of Vahlen's group, i.e. a group of $2$-by-$2$ matrices with entries in the Clifford algebra satisfying certain conditions....
Topics: Group Theory, Rings and Algebras, Mathematics
Source: http://arxiv.org/abs/1609.08308
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Jun 29, 2018
06/18
by
Ralf Schiffler; Khrystyna Serhiyenko
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Every cluster-tilted algebra $B$ is the relation extension $C\ltimes \text{Ext}^2_C(DC,C)$ of a tilted algebra $C$. A $B$-module is called induced if it is of the form $M\otimes_C B$ for some $C$-module $M$. We study the relation between the injective presentations of a $C$-module and the injective presentations of the induced $B$-module. Our main result is an explicit construction of the modules and morphisms in an injective presentation of any induced $B$-module. In the case where the...
Topics: Representation Theory, Rings and Algebras, Mathematics
Source: http://arxiv.org/abs/1604.06907
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Jun 28, 2018
06/18
by
Victor Kozyakin
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Recently Blondel, Nesterov and Protasov proved that the finiteness conjecture holds for the generalized and the lower spectral radii of the sets of non-negative matrices with independent row/column uncertainty. We show that this result can be obtained as a simple consequence of the so-called hourglass alternative earlier used by the author and his companions to analyze the minimax relations between the spectral radii of matrix products. Axiomatization of the statements that constitute the...
Topics: Mathematics, Rings and Algebras
Source: http://arxiv.org/abs/1507.00492
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Jun 28, 2018
06/18
by
Georgi Dimov; Dimiter Vakarelov
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The results of Iv. Prodanov on abstract spectra and separative algebras were announced in the journal "Trudy Mat. Inst. Steklova", 154, 1983, 200--208, but their proofs were never written by him in the form of a manuscript, preprint or paper. Since the untimely death of Ivan Prodanov withheld him from preparing the full version of this paper and since, in our opinion, it contains interesting and important results, we undertook the task of writing a full version of it and thus making...
Topics: Mathematics, Category Theory, Rings and Algebras, General Topology
Source: http://arxiv.org/abs/1507.05486
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Jun 30, 2018
06/18
by
Dmitry Millionschikov
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We study the algebraic constraints on the structure of nilpotent Lie algebra $\mathbb{g}$, which arise because of the presence of an integrable complex structure $J$. Particular attention is paid to non-abelian complex structures. Constructed various examples of positive graded Lie algebras with complex structures, in particular, we construct an infinite family $\mathfrak{D}(n)$ of such algebras that we have for their nil-index $s(\mathfrak{D}(n))$: $$ s(\mathfrak{D}(n))=[...
Topics: Mathematics, Rings and Algebras
Source: http://arxiv.org/abs/1412.0361
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Jun 30, 2018
06/18
by
Reinhold Burger; Albert Heinle
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In this paper we present a new primitive for a key exchange protocol based on multivariate non-commutative polynomial rings, analogous to the classic Diffie-Hellman method. Our technique extends the proposed scheme of Boucher et al. from 2010. Their method was broken by Dubois and Kammerer in 2011, who exploited the Euclidean domain structure of the chosen ring. However, our proposal is immune against such attacks, without losing the advantages of non-commutative polynomial rings as outlined by...
Topics: Cryptography and Security, Mathematics, Symbolic Computation, Rings and Algebras, Computing...
Source: http://arxiv.org/abs/1407.1270
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Jun 30, 2018
06/18
by
Jinyong Wu; Gongxiang Liu; Nanqing Ding
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The classification of affine prime regular Hopf algebras of GK-dimension one is completed. As consequences, 1) we give a negative answer to an open question posed by Brown-Zhang and 2) we show that there do exist prime regular Hopf algebras of GK-dimension one which are not pointed.
Topics: Mathematics, Quantum Algebra, Rings and Algebras
Source: http://arxiv.org/abs/1410.7497
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Jun 30, 2018
06/18
by
Geoffrey Mason; Siu-Hung Ng
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Given a pair of finite groups $F, G$ and a normalized 3-cocycle $\omega$ of $G$, where $F$ acts on $G$ as automorphisms, we consider quasi-Hopf algebras defined as a cleft extension $\Bbbk^G_\omega\#_c\,\Bbbk F$ where $c$ denotes some suitable cohomological data. When $F\rightarrow \overline{F}:=F/A$ is a quotient of $F$ by a central subgroup $A$ acting trivially on $G$, we give necessary and sufficient conditions for the existence of a surjection of quasi-Hopf algebras and cleft extensions of...
Topics: High Energy Physics - Theory, Group Theory, Quantum Algebra, Mathematics, Representation Theory,...
Source: http://arxiv.org/abs/1404.2016
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Jun 30, 2018
06/18
by
Sebastian Herpel; David I. Stewart
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We provide results on the smoothness of normalisers in connected reductive algebraic groups $G$ over fields $k$ of positive characteristic $p$. Specifically we we give bounds on $p$ which guarantee that normalisers of subalgebras of $\mathfrak{g}$ in $G$ are smooth, i.e.\ so that the Lie algebras of these normalisers coincide with the infinitesimal normalisers. One of our main tools is to exploit cohomology vanishing of small dimensional modules. Along the way, we obtain complete reducibility...
Topics: Mathematics, Rings and Algebras, Representation Theory, Group Theory
Source: http://arxiv.org/abs/1402.6280
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Jun 30, 2018
06/18
by
Frauke M. Bleher; Ted Chinburg; Birge Huisgen-Zimmermann
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Let $\Lambda$ be a basic finite dimensional algebra over an algebraically closed field, with the property that the square of the Jacobson radical $J$ vanishes. We determine the irreducible components of the module variety $\text{Mod}_{\bf d}(\Lambda)$ for any dimension vector $\bf d$. Our description leads to a count of the components in terms of the underlying Gabriel quiver. A closed formula for the number of components when $\Lambda$ is local extends existing counts for the two-loop quiver...
Topics: Mathematics, Rings and Algebras, Representation Theory
Source: http://arxiv.org/abs/1407.3045
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Jun 26, 2018
06/18
by
Igor Buchberger; Jürgen Fuchs
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As a step towards the structure theory of Lie algebras in symmetric monoidal categories we establish results involving the Killing form. The proper categorical setting for discussing these issues are symmetric ribbon categories.
Topics: Mathematics, Category Theory, Representation Theory, Rings and Algebras
Source: http://arxiv.org/abs/1502.07441
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Jun 27, 2018
06/18
by
Fernando Szechtman
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Let $f(Z)=Z^n-a_{1}Z^{n-1}+\cdots+(-1)^{n-1}a_{n-1}Z+(-1)^na_n$ be a monic polynomial with coefficients in a ring~$R$ with identity, not necessarily commutative. We study the ideal $I_f$ of $R[X_1,\dots,X_n]$ generated by $\sigma_i(X_1,\dots,X_n)-a_{i}$, where $\sigma_1,\dots,\sigma_n$ are the elementary symmetric polynomials, as well as the quotient ring $R[X_1,\dots,X_n]/I_f$.
Topics: Rings and Algebras, Mathematics
Source: http://arxiv.org/abs/1504.00973
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Jun 27, 2018
06/18
by
Jesse Elliott
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We show that the theory of quantales and quantic nuclei motivate new results on star operations, semistar operations, semiprime operations, ideal systems, and module systems, and conversely the latter theories motivate new results on quantales and quantic nuclei. Results include representation theorems for precoherent prequantales and multiplicative semilattices; characterizations of the simple prequantales; and a generalization to the setting of precoherent quantales of the construction of the...
Topics: Mathematics, Rings and Algebras
Source: http://arxiv.org/abs/1505.06433
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Jun 26, 2018
06/18
by
Kira Adaricheva; J. B. Nation
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Various characterizations of finite convex geometries are well known. This note provides similar characterizations for possibly infinite convex geometries whose lattice of closed sets is strongly coatomic and lower continuous. Some classes of examples of such convex geometries are given.
Topics: Mathematics, Combinatorics, Rings and Algebras
Source: http://arxiv.org/abs/1501.04174
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Jun 30, 2018
06/18
by
Neville Fogarty; Heide Gluesing-Luerssen
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We introduce circulant matrices that capture the structure of a skew-polynomial ring F[x;\theta] modulo the left ideal generated by a polynomial of the type x^n-a. This allows us to develop an approach to skew-constacyclic codes based on such circulants. Properties of these circulants are derived, and in particular it is shown that the transpose of a certain circulant is a circulant again. This recovers the well-known result that the dual of a skew-constacyclic code is a constacyclic code...
Topics: Mathematics, Rings and Algebras, Computing Research Repository, Information Theory
Source: http://arxiv.org/abs/1408.5445
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Jun 30, 2018
06/18
by
Christopher Davis; Tommy Occhipinti
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We prove that if $G$ is a finite simple group which is the unit group of a ring, then $G$ is isomorphic to either (a) a cyclic group of order 2; (b) a cyclic group of prime order $2^k -1$ for some $k$; or (c) a projective special linear group $PSL_n(\mathbb{F}_2)$ for some $n \geq 3$. Moreover, these groups do (trivially) all occur as unit groups. We deduce this classification from a more general result, which holds for groups $G$ with no non-trivial normal 2-subgroup.
Topics: Mathematics, Rings and Algebras, Group Theory
Source: http://arxiv.org/abs/1409.7518
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Jun 30, 2018
06/18
by
Ramon Antoine; Francesc Perera; Hannes Thiel
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The Cuntz semigroup of a C*-algebra is an important invariant in the structure and classification theory of C*-algebras. It captures more information than K-theory but is often more delicate to handle. We systematically study the lattice and category theoretic aspects of Cuntz semigroups. Given a C*-algebra $A$, its (concrete) Cuntz semigroup $Cu(A)$ is an object in the category $Cu$ of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu. To clarify the distinction...
Topics: Mathematics, Category Theory, Rings and Algebras, Operator Algebras
Source: http://arxiv.org/abs/1410.0483
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Jun 29, 2018
06/18
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Paul Martin; Volodymyr Mazorchuk
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Starting from the symmetric group $S_n$, we construct two fiat $2$-categories. One of them can be viewed as the fiat "extension" of the natural $2$-category associated with the symmetric inverse semigroup (considered as an ordered semigroup with respect to the natural order). This $2$-category provides a fiat categorification for the integral semigroup algebra of the symmetric inverse semigroup. The other $2$-category can be viewed as the fiat "extension" of the $2$-category...
Topics: Category Theory, Group Theory, Representation Theory, Rings and Algebras, Mathematics
Source: http://arxiv.org/abs/1605.03880
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Jun 30, 2018
06/18
by
Mehmet Uc; Mustafa Alkan
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Let R be a commutative ring with unity, M a module over R and let S be a G-set for a finite group G. We define a set MS to be the set of elements expressed as the formal finite sum of the form similar to the elements of group ring RG. The set MS is a module over the group ring RG under the addition and the scalar multiplication similar to the RG-module MG. With this notion, we not only generalize but also unify the theories of both of the group algebra and the group module, and we also...
Topics: Rings and Algebras, Commutative Algebra, Mathematics
Source: http://arxiv.org/abs/1701.06444
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Jun 30, 2018
06/18
by
Jörg Feldvoss; Salvatore Siciliano; Thomas Weigel
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In this paper we show that the projective cover of the trivial irreducible module of a finite-dimensional solvable restricted Lie algebra is induced from the one-dimensional trivial module of a maximal torus. As a consequence, we obtain that the number of the isomorphism classes of irreducible modules with a fixed p-character for a finite-dimensional solvable restricted Lie algebra L is bounded above by p^MT(L), where MT(L) denotes the largest dimension of a torus in L. Finally, we prove that...
Topics: Mathematics, Rings and Algebras, Representation Theory
Source: http://arxiv.org/abs/1407.1902
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Jun 28, 2018
06/18
by
Adel Alahmadi; Hamed Alsulami
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We prove that if $A$ is finitely presented algebra with idempotent $e$ such that $A=AeA=A(1-e)A$ then the algebra $eAe$ is finitely presented.
Topics: Mathematics, Rings and Algebras
Source: http://arxiv.org/abs/1507.07431
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Jun 29, 2018
06/18
by
Pilar Benito; Daniel de-la-Concepción; Jesús Laliena
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In this paper we introduce an equivalence between the category of the t-nilpotent quadratic Lie algebras with d generators and the category of some symmetric invariant bilinear forms on the t-nilpotent free Lie algebra with d generators. Taking into account this equivalence, t-nilpotent quadratic Lie algebras with d generators are classified (up to isometric isomorphism, and over any field of characteristic zero), in the following cases: d=2 and t
Topics: Rings and Algebras, Mathematics
Source: http://arxiv.org/abs/1604.02923
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Jun 27, 2018
06/18
by
Hop D. Nguyen; Thanh Vu
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This work concerns the linearity defect of a module $M$ over a noetherian local ring $R$, introduced by Herzog and Iyengar in 2005, and denoted by $\text{ld}_R M$. Roughly speaking, $\text{ld}_R M$ is the homological degree beyond which the minimal free resolution of $M$ is linear. In the paper, it is proved that for any ideal $I$ in a regular local ring $R$ and for any finitely generated $R$-module $M$, each of the sequences $(\text{ld}_R (I^nM))_n$ and $(\text{ld}_R (M/I^nM))_n$ is eventually...
Topics: Commutative Algebra, Mathematics, Rings and Algebras
Source: http://arxiv.org/abs/1504.04853
