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आयतन 5, मुद्दा 1 (2011)

समीक्षा लेख

Contractions of 3-Dimensional Representations of the Lie Algebra sl(2)

Jan Smotlacha and Goce Chadzitaskos

A theory of grading preserving contractions of representations of Lie algebras has been developed. In this theory, grading of the given Lie algebra is characterized by two sets of parameters satisfying a derived set of equations. Here we introduce a list of resulting 3-dimensional representations for the Z3-grading of the sl(2) Lie algebra.

संपादकीय

Preface to the Special Issue on Deformation Theory and Applications

Abdenacer Makhlouf,Eugen Paal,and Alexander Stolin

The purpose of this issue is to present some contributions that develop deformations of algebraic structures and applications to physics and their interrelations. It shows how basic techniques of deformation theory work in various standard situations.

शोध आलेख

The Generalized Burnside Theorem in Noncommutative Deformation Theory

Eivind Eriksen

Let A be an associative algebra over a field k, and letMbe a finite family of right A-modules. A study of the noncommutative deformation functor DefM of the familyMleads to the construction of the algebra OA(M) of observables and the generalized Burnside theorem, due to Laudal (2002). In this paper, we give an overview of aspects of noncommutative deformations closely connected to the generalized Burnside theorem.

शोध आलेख

Generalized Matric Massey Products for Graded Modules

Arvid Siqveland

The theory of generalized matric Massey products has been applied for some time to A-modules M, A being a k-algebra. The main application is to compute the local formal moduli ˆHM, isomorphic to the local ring of the moduli of A-modules. This theory is also generalized to OX-modulesM, X being a k-scheme. In these notes, we consider the definition of generalized Massey products and the relation algebra in any obstruction situation (a differential graded k-algebra with certain properties), and prove that this theory applies to the case of graded Rmodules, R being a graded k-algebra and k algebraically closed. When the relation algebra is algebraizable, that is, the relations are polynomials rather than power series, this gives a combinatorial way to compute open (´etale) subsets of the moduli of graded R-modules. This also gives a sufficient condition for the corresponding point in the moduli of O Proj(R)-modules to be singular. The computations are straightforwardly algorithmic, and an example on the postulation Hilbert scheme is given.

शोध आलेख

Geometry of Noncommutative k-Algebras

Arvid Siqveland

Let X be a scheme over an algebraically closed field k, and let x ∈ SpecR ⊆ X be a closed point corresponding to the maximal ideal m ⊆ R. Then OˆX,x is isomorphic to the prorepresenting hull, or local formal moduli, of the deformation functor DefR/m :  → Sets. This suffices to reconstruct X up to etal´e coverings. For a noncommutative k-algebra A the simple modules are not necessarily of dimension one, and there is a geometry between them. We replace the points in the commutative situation with finite families of points in the noncommutative situation, and replace the geometry of points with the geometry of sets of points given by noncommutative deformation theory. We apply the theory to the noncommutative moduli of three-dimensional endomorphisms.

शोध आलेख

Dynamical Yang-Baxter Maps Associated with Homogeneous Pre-Systems

Noriaki Kamiya and Youichi Shibukawa

We construct dynamical Yang-Baxter maps, which are set-theoretical solutions to a version of the quantum dynamical Yang-Baxter equation, by means of homogeneous pre-systems, that is, ternary systems encoded in the reductive homogeneous space satisfying suitable conditions. Moreover, a characterization of these dynamical Yang- Baxter maps is presented.

शोध आलेख

Cohomology and Formal Deformations of Alternative Algebras

Mohamed Elhamdadi1 and Abdenacer Makhlouf

The purpose of this paper is to introduce an algebraic cohomology and formal deformation theory of alternative algebras. A short review of basics on alternative algebras and their connections to some other algebraic structures is also provided.

शोध आलेख

Phase Spaces and Deformation Theory

Olav Arnfinn Laudal

We have previously introduced the notion of non-commutative phase space (algebra) associated to any associative algebra, defined over a field. The purpose of the present paper is to prove that this construction is useful in non-commutative deformation theory for the construction of the versal family of finite families of modules. In particular, we obtain a much better understanding of the obstruction calculus, that is, of the Massey products.

शोध आलेख

Meanders and Frobenius Seaweed Lie Algebras

Vincent Coll,Anthony Giaquinto,and Colton Magnant

The index of a seaweed Lie algebra can be computed from its associated meander graph.We examine this graph in several ways with a goal of determining families of Frobenius (index zero) seaweed algebras. Our analysis gives two new families of Frobenius seaweed algebras as well as elementary proofs of known families of such Lie algebras.

शोध आलेख

Deformations of Complex 3-Dimensional Associative Algebras

Alice Fialowski,Michael Penkava,and Mitch Phillipson

We study deformations and the moduli space of 3-dimensional complex associative algebras. We use extensions to compute the moduli space, and then give a decomposition of this moduli space into strata consisting of complex projective orbifolds, glued together through jump deformations. The main purpose of this paper is to give a logically organized description of the moduli space, and to give an explicit description of how the moduli space is constructed by extensions.

शोध आलेख

On Graded Global Dimension of Color Hopf Algebras

Yan-HuaWang

In this paper, we prove the fundamental theorem of color Hopf module similar to the fundamental theorem of Hopf module. As an application, we prove that the graded global dimension of a color Hopf algebra coincides with the projective dimension of the trivial module K.

शोध आलेख

Some Properties of Intuitionistic Fuzzy Lie Algebras over a Fuzzy Field

P.L.Antony and P.L.Lilly

The concept of intuitionistic fuzzy Lie algebra over a fuzzy field is introduced. We study the “necessity” and “possibility” operators on intuitionistic fuzzy Lie algebra over a fuzzy field and give some properties of homomorphic images.

शोध आलेख

An Approach to Omni-Lie Algebroids Using Quasi-Derivations

We introduce the notion of left (and right) quasi-Loday algebroids and a “universal space” for them, called a left (right) omni-Loday algebroid, in such a way that Lie algebroids, omni-Lie algebras and omni-Loday algebroids are particular substructures.

शोध आलेख

Algebraic Structures Derived from Foams

J. Scott Carter1 and Masahico Saito

Foams are surfaces with branch lines at which three sheets merge. They have been used in the categorification of sl(3) quantum knot invariants and also in physics. The 2D-TQFT of surfaces, on the other hand, is classified by means of commutative Frobenius algebras, where saddle points correspond to multiplication and comultiplication. In this paper, we explore algebraic operations that branch lines derive under TQFT. In particular, we investigate Lie bracket and bialgebra structures. Relations to the original Frobenius algebra structures are discussed both algebraically and diagrammatically.

शोध आलेख

Stability of the Generalized Polar Decomposition Method for the Approximation of the Matrix Exponential

Elham Nobari and S. Mohammad Hosseini

Generalized polar decomposition method (or briefly GPD method) has been introduced by Munthe-Kaas and Zanna [5] to approximate the matrix exponential. In this paper, we investigate the numerical stability of that method with respect to roundoff propagation. The numerical GPD method includes two parts: splitting of a matrix Z ∈ g, a Lie algebra of matrices and computing exp(Z)v for a vector v. We show that the former is stable provided that Z is not so large, while the latter is not stable in general except with some restrictions on the entries of the matrix Z and the vector v.

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