Showing papers on "Algebra representation published in 2012"
01 Jan 2012
TL;DR: A compilation of two sets of notes at the University of Kansas was published in the Spring of 2002 by?? and the other in the spring of 2007 by Branden Stone.
Abstract: 1 A compilation of two sets of notes at the University of Kansas; one in the Spring of 2002 by ?? and the other in the Spring of 2007 by Branden Stone. These notes have been typed
1,289 citations
Book•
10 Mar 2012
TL;DR: The book gives a thorough introduction to the mathematical underpinnings of computer algebra, and in bridging the gap between the algebraic theory and computer algebra software, should be of interest to both mathematics and computer science students.
Abstract: The book gives a thorough introduction to the mathematical underpinnings of computer algebra. The subjects treated range from arithmetic of integers and polynomials to fast factorization methods, Grobner bases, and algorithms in algebraic geometry. The algebraic background for all the algorithms presented in the book is fully described, and most of the algorithms are investigated with respect to their computational complexity. Each chapter closes with a brief survey of the related literature. The book is designed as a textbook for a course in computer algebra for advanced undergraduate or beginning graduate students. Every chapter contains a considerable number of exercises, some of which are solved in the appendix. In bridging the gap between the algebraic theory and computer algebra software, the book should be of interest to both mathematics and computer science students.
274 citations
TL;DR: In this article, the authors provide a spacetime interpretation in terms of a novel contraction of the parent algebra, and discuss implications for the BMS/GCA correspondence and the consequences of this contraction on the boundary conformal field theory.
Abstract: The asymptotic group of symmetries at null infinity of flat spacetimes in three and four dimensions is the infinite dimensional Bondi-Metzner-Sachs (BMS) group. This has recently been shown to be isomorphic to non-relativistic conformal algebras in one lower dimension, the Galilean Conformal Algebra (GCA) in 2d and a closely related non-relativistic algebra in 3d [1, 2]. We provide a better understanding of this surprising connection by providing a spacetime interpretation in terms of a novel contraction. The 2d GCA was previously obtained from a linear combination of two copies of the Virasoro algebra. We consider a representation obtained from a different linear combination of the Virasoros, which is relevant to the relation with the BMS algebra in three dimensions. This is realised by a new space-time contraction of the parent algebra. We show that this representation has interesting correlation functions. We discuss implications for the BMS/GCA correspondence and show that the flat space limit actually induces precisely this contraction on the boundary conformal field theory. We also discuss aspects of asymptotic symmetries and the consequences of this contraction in higher dimensions.
246 citations
TL;DR: In this article, the equivalence between the refined topological vertex of Iqbal-Kozcaz-Vafa and a certain representation theory of the quantum algebra of type W 1+∞ introduced by Miki was established.
Abstract: We establish the equivalence between the refined topological vertex of Iqbal-Kozcaz-Vafa and a certain representation theory of the quantum algebra of type W
1+∞ introduced by Miki. Our construction involves trivalent intertwining operators Φ and Φ* associated with triples of the bosonic Fock modules. Resembling the topological vertex, a triple of vectors ∈ $$ {\mathbb{Z}^2} $$
is attached to each intertwining operator, which satisfy the Calabi-Yau and smoothness conditions. It is shown that certain matrix elements of Φ and Φ* give the refined topological vertex C
λμν
(t, q) of Iqbal-Kozcaz-Vafa. With another choice of basis, we recover the refined topological vertex C
λμ
ν
(q, t) of Awata-Kanno. The gluing factors appears correctly when we consider any compositions of Φ and Φ*. The spectral parameters attached to Fock spaces play the role of the Kahler parameters.
138 citations
TL;DR: In this paper, a correspondence between Young diagrams and differential operators of infinitely many variables is established, and the Schur functions form a complete system of common eigenfunctions of these differential operators, and their eigenvalues are expressed through the characters of symmetric groups.
136 citations
TL;DR: Aspects of the D = 4, \({\mathcal N=4}\) superconformal symmetry relevant to the AdS/CFT duality and integrability are reviewed in this article.
Abstract: Aspects of the D = 4, \({\mathcal N=4}\) superconformal symmetry relevant to the AdS/CFT duality and integrability are reviewed. These include the Lie superalgebra \({\mathfrak {psu}(2,2|4)}\), its representations, conformal transformations and correlation functions in \({\mathcal N=4}\) super Yang–Mills theory as well as an illustration of the AdS5 × S5 superspace on which the dual string theory is formulated.
99 citations
TL;DR: In this paper, the authors complete the Schwinger Dyson equations and the associated algebra at all orders in 1 / N, where the full algebra of constraints is indexed by D-colored graphs, and the leading order D-ary tree algebra is a Lie subalgebra of the full constraints algebra.
93 citations
Posted Content•
TL;DR: The theory of twisted commutative algebras can be seen as a generalization of the theory of linear symmetries as mentioned in this paper, which can be thought of as a theory for handling large groups of symmetric groups.
Abstract: This article is an expository account of the theory of twisted commutative algebras, which simply put, can be thought of as a theory for handling commutative algebras with large groups of linear symmetries. Examples include the coordinate rings of determinantal varieties, Segre-Veronese embeddings, and Grassmannians. The article is meant to serve as a gentle introduction to the papers of the two authors on the subject, and also to point out some literature in which these algebras appear. The first part reviews the representation theory of the symmetric groups and general linear groups. The second part introduces a related category and develops its basic properties. The third part develops some basic properties of twisted commutative algebras from the perspective of classical commutative algebra and summarizes some of the results of the authors. We have tried to keep the prerequisites to this article at a minimum. The article is aimed at graduate students interested in commutative algebra, algebraic combinatorics, or representation theory, and the interactions between these subjects.
89 citations
TL;DR: In this article, the authors prove the rationality of all the minimal series principal W-algebras discovered by Frenkel, Kac and Wakimoto in 1992, thereby giving a new family of rational and C_2-cofinite vertex operator algebrAs.
Abstract: We prove the rationality of all the minimal series principal W-algebras discovered by Frenkel, Kac and Wakimoto in 1992, thereby giving a new family of rational and C_2-cofinite vertex operator algebras. A key ingredient in our proof is the study of Zhu's algebra of simple W-algebras via the quantized Drinfeld-Sokolov reduction. We show that the functor of taking Zhu's algebra commutes with the reduction functor. Using this general fact we determine the maximal spectrums of the associated graded of Zhu's algebra of all the admissible affine vertex algebras as well.
80 citations
Posted Content•
TL;DR: In this article, the authors studied properties of the central sequence algebra of a C*-algebra and presented an alternative approach to a recent result of Matui and Sato, which they extended to the case where the extreme boundary of the trace simplex is closed and of finite topological dimension.
Abstract: We study properties of the central sequence algebra of a C*-algebra, and we present an alternative approach to a recent result of Matui and Sato. They prove that every unital separable simple nuclear C*-algebra, whose trace simplex is finite dimensional, tensorially absorbs the Jiang-Su algebra if and only if it has the strict comparison property. We extend their result to the case where the extreme boundary of the trace simplex is closed and of finite topological dimension. We are also able to relax the assumption on the C*-algebra of having the strict comparison property to a weaker property, that we call local weak comparison. Namely, we prove that a unital separable simple nuclear C*-algebra, whose trace simplex has finite dimensional closed extreme boundary, tensorially absorbs the Jiang-Su algebra if and only if it has the local weak comparison property.
We can also eliminate the nuclearity assumption, if instead we assume the (SI) property of Matui and Sato, and, moreover, that each II_1-factor representation of the C*-algebra is a McDuff factor.
Posted Content•
TL;DR: In this article, the authors introduce new features of the shuffle algebra that will allow them to obtain explicit formulas for the isomorphism between its Drinfeld double and the elliptic Hall algebra.
Abstract: In this paper, we introduce certain new features of the shuffle algebra, that will allow us to obtain explicit formulas for the isomorphism between its Drinfeld double and the elliptic Hall algebra.
TL;DR: In this article, it was shown that the category of n -ary Hom-Nambu(−Lie) algebras is closed under twisting by self-weak morphisms.
TL;DR: In this paper, the authors extend the framework of cellular algebras to affine cellular algebra and affine Hecke algebra, which need not be finite dimensional over a field.
TL;DR: In this paper, the Bose condensate ground states have been constructed for the Lie algebra as well as the Weyl algebra setting, and discussed possible applications in effective field theory, loop quantum cosmology, and further generalizations.
Abstract: In loop quantum gravity, states of the gravitational field turn out to be excita- tions over a vacuum state that is sharply peaked on a degenerate spatial geometry. While this vacuum is singled out as fundamental due to its invariance properties, it is also impor- tant to consider states that describe non-degenerate geometries. Such states have features of Bose condensate ground states. We discuss their construction for the Lie algebra as well as the Weyl algebra setting, and point out possible applications in effective field theory, Loop Quantum Cosmology, as well as further generalizations.
Posted Content•
TL;DR: In this paper, it was shown that if the trace space of A has compact finite-dimensional extreme boundary, then there exist unital embeddings of matrix algebras into a certain central sequence algebra of A.
Abstract: Let A be a unital separable simple infinite-dimensional nuclear C*-algebra with at least one tracial state. We prove that if the trace space of A has compact finite-dimensional extreme boundary then there exist unital embeddings of matrix algebras into a certain central sequence algebra of A which is determined by the uniform topology on the trace space. As an application, it is shown that if furthermore A has strict comparison then A absorbs the Jiang-Su algebra tensorially.
TL;DR: In this article, a generalized description for the κ-Poincare-Hopf algebra as a symmetry quantum group of underlying Minkowski spacetime is proposed, and a three-parameter family of deformed Lorentz generators with κ Poincare algebras is constructed.
TL;DR: In this article, an analytic continuation of the twisted representation of the Heisenberg vertex algebra F was constructed using period integrals, which yields a global object, which may be called a W-twisted representation of F. The main result is that the total descendant potential of the singularity is a highest weight vector for the W-algebra.
Abstract: Simple, or Kleinian, singularities are classified by Dynkin diagrams of type ADE. Let g be the corresponding finite-dimensional Lie algebra, and W its Weyl group. The set of g-invariants in the basic representation of the affine Kac-Moody algebra g^ is known as a W-algebra and is a subalgebra of the Heisenberg vertex algebra F. Using period integrals, we construct an analytic continuation of the twisted representation of F. Our construction yields a global object, which may be called a W-twisted representation of F. Our main result is that the total descendant potential of the singularity, introduced by Givental, is a highest weight vector for the W-algebra.
Posted Content•
TL;DR: In this paper, Minamoto's theory on Fano algebras in non-commutative algebraic geometry is applied to n-representation infinite modules, which are a class of finite-dimensional finite-diagramal algebraic modules of global dimension n. They are a certain analog of representation infinite hereditary algesbras.
Abstract: From the viewpoint of higher dimensional Auslander-Reiten theory, we introduce a new class of finite dimensional algebras of global dimension n, which we call n-representation infinite. They are a certain analog of representation infinite hereditary algebras, and we study three important classes of modules: n-preprojective, n-preinjective and n-regular modules. We observe that their homological behaviour is quite interesting. For instance they provide first examples of algebras having infinite Ext^1-orthogonal families of modules. Moreover we give general constructions of n-representation infinite algebras.
Applying Minamoto's theory on Fano algebras in non-commutative algebraic geometry, we describe the category of n-regular modules in terms of the corresponding preprojective algebra. Then we introduce n-representation tame algebras, and show that the category of n-regular modules decomposes into the categories of finite dimensional modules over localizations of the preprojective algebra. This generalizes the classical description of regular modules over tame hereditary algebras. As an application, we show that the representation dimension of an n-representation tame algebra is at least n+2.
TL;DR: In this paper, the authors introduce the concept of an extended O-operator on an associative algebra whose Lie algebra analog has been applied to generalized Lax pairs and PostLie algebras.
Abstract: An O-operator on an associative algebra is a generalization of a Rota-Baxter operator that plays an important role in the Hopf algebra approach of Connes and Kreimer to the renormalization of quantum field theory. It is also the associative analog of an O-operator on a Lie algebra in the study of the classical Yang-Baxter equation. We introduce the concept of an extended O-operator on an associative algebra whose Lie algebra analog has been applied to generalized Lax pairs and PostLie algebras. We study algebraic structures coming from extended O-operators. Continuing the work of Aguiar deriving Rota-Baxter operators from the associative Yang-Baxter equation, we show that its solutions correspond to extended O-operators through a duality. We also establish a relationship of extended O-operators with the generalized associative Yang-Baxter equation.
TL;DR: In this article, an unconventional realization of the Poincare algebra of special relativity as conformal transformations was introduced, and it may occur as a dynamical symmetry algebra for ageing systems in non-equilibrium statistical physics.
Abstract: By introducing an unconventional realization of the Poincare algebra of special relativity as conformal transformations, we show how it may occur as a dynamical symmetry algebra for ageing systems in non-equilibrium statistical physics and give some applications, such as the computation of two-time correlators. We also discuss infinite-dimensional extensions of in this setting. Finally, we construct canonical Appell systems, coherent states and Leibniz function for as a tool for bosonic quantization.
TL;DR: In this article, the authors extend the definition of local Weyl modules to the setting of equivariant map algebras where g is semisimple, X is affine of finite type, and the group is abelian.
TL;DR: In this article, it was shown that nuclear C ∗ -algebras have a refined version of the completely positive approximation property, in which the maps that approximately factorize through finite dimensional algebrains are convex combinations of order zero maps.
Posted Content•
TL;DR: In this paper, Zhu's algebra and C_2-algebra of parafermion vertex operator algebras for sl_2 are shown to be C 2-cofiniteness.
Abstract: We determine Zhu's algebra and C_2-algebra of parafermion vertex operator algebras for sl_2. Moreover, we prove the C_2-cofiniteness of parafermion vertex operator algebras for any finite dimensional simple Lie algebras.
Book•
13 Sep 2012TL;DR: In this paper, the authors present a self-contained overview of the classical and current aspects of deformation theory, assuming only basic knowledge of commutative algebra, homological algebra and category theory.
Abstract: This book brings together both the classical and current aspects of deformation theory. The presentation is mostly self-contained, assuming only basic knowledge of commutative algebra, homological algebra and category theory. In the interest of readability, some technically complicated proofs have been omitted when a suitable reference was available. The relation between the uniform continuity of algebraic maps and topologized tensor products is explained in detail, however, as this subject does not seem to be commonly known and the literature is scarce. The exposition begins by recalling Gerstenhaber's classical theory for associative algebras. The focus then shifts to a homotopy-invariant setup of Maurer-Cartan moduli spaces. As an application, Kontsevich's approach to deformation quantization of Poisson manifolds is reviewed. Then, after a brief introduction to operads, a strongly homotopy Lie algebra governing deformations of (diagrams of) algebras of a given type is described, followed by examples and generalizations.
TL;DR: In this article, the Schur elements associated to the simple modules of the Ariki-Koike algebra were studied and a cancellation-free formula for them was given so that their factors can be easily read and programmed.
Abstract: We study the Schur elements associated to the simple modules of the Ariki---Koike algebra. We first give a cancellation-free formula for them so that their factors can be easily read and programmed. We then study direct applications of this result. We also complete the determination of the canonical basic sets for cyclotomic Hecke algebras of type G(l,p,n) in characteristic 0.
TL;DR: This paper characterizations the class of finite idempotent algebras having cube-terms and yields a polynomial-time algorithm for determining if the algebra has a cube-term, and determines the maximal non-finitely related idempotsent clones over A.
Abstract: Aichinger et al. (2011) have proved that every finite algebra with a cube-term (equivalently, with a parallelogram-term; equivalently, having few subpowers) is finitely related. Thus finite algebras with cube terms are inherently finitely related—every expansion of the algebra by adding more operations is finitely related. In this paper, we show that conversely, if A is a finite idempotent algebra and every idempotent expansion of A is finitely related, then A has a cube-term. We present further characterizations of the class of finite idempotent algebras having cube-terms, one of which yields, for idempotent algebras with finitely many basic operations and a fixed finite universe A, a polynomial-time algorithm for determining if the algebra has a cube-term. We also determine the maximal non-finitely related idempotent clones over A. The number of these clones is finite.
TL;DR: In this article, the authors define the notion of action of an L-infinity algebra on a graded manifold and show that such an action corresponds to a homological vector field on the manifold.
Abstract: We define the notion of action of an L-infinity algebra $g$ on a graded manifold $M$, and show that such an action corresponds to a homological vector field on $g[1] \times M$ of a specific form. This generalizes the correspondence between Lie algebra actions on manifolds and transformation Lie algebroids. In particular, we consider actions of $g$ on a second L-infinity algebra, leading to a notion of "semidirect product" of L-infinity algebras more general than those we found in the literature.