Journal of Generalized Lie Theory and Applications Articles (Project Euclid)

Notes on cohomologies of ternary algebras of associative type

Fri, 06 Aug 2010 10:55 EDT

H. Ataguema, A. Makhlouf

Source: J. Gen. Lie Theory Appl., Volume 3, Number 3, 157--174.

Abstract:
The aim of this paper is to investigate the cohomologies for ternary algebras of associative type. We study in particular the cases of partially associative ternary algebras and weak totally associative ternary algebras. Also, we consider the Takhtajan's construction, which was used to construct a cohomology of ternary Nambu-Lie algebras using Chevalley-Eilenberg cohomology of Lie algebras, and discuss it in the case of ternary algebras of associative type. One of the main results of this paper states that a usual deformation cohomology does not exist for partially associative ternary algebras which implies that their operad is not a Koszul operad.

A special form of Rund's h-curvature tensor using $R3$-like Finsler space

Fri, 06 Aug 2010 10:55 EDT

S. T. Aveesh, S. K. Narasimhamurthy, H. G. Nagaraja, Pradeep Kumar

Source: J. Gen. Lie Theory Appl., Volume 3, Number 3, 175--180.

Abstract:
The purpose of the present paper is to consider and study a special form of Rund's h-curvature tensor $K^i_{ljk}$ and Berwald's curvature tensor $H^i_{ljk}$ in an $R3$-like $C$-reducible Finsler space. In this paper, we modify the Rund's h-curvature tensor $K^i_{ljk}$ to special form by using some special Finsler spaces like $C$-reducible, $R3$-like Finsler spaces.

Log-concavity of the cohomology of nilpotent Lie algebras in characteristic two

Fri, 06 Aug 2010 10:55 EDT

Grant Cairns

Source: J. Gen. Lie Theory Appl., Volume 3, Number 3, 181--182.

Abstract:
It is known that the Betti numbers of the Heisenberg Lie algebras are unimodal over fields of characteristic two. This note observes that they are log-concave. An example is given of a nilpotent Lie algebra in characteristic two for which the Betti numbers are unimodal but not log-concave.

On anti-structurable algebras and extended Dynkin diagrams

Fri, 06 Aug 2010 10:55 EDT

Noriaki Kamiya, Daniel Mondoc

Source: J. Gen. Lie Theory Appl., Volume 3, Number 3, 183--190.

Abstract:
We construct Lie superalgebras $\mathfrak{osp}(2n+1|4n+2)$ and $\mathfrak{osp}(2n|4n)$ starting with certain classes of anti-structurable algebras via the standard embedding Lie superalgebra construction corresponding to (ε,δ)-Freudenthal Kantor triple systems.

Bruck decomposition for endomorphisms of quasigroups

Fri, 06 Aug 2010 10:55 EDT

Péter T. Nagy, Peter Plaumann

Source: J. Gen. Lie Theory Appl., Volume 3, Number 3, 191--196.

Abstract:
In 1944, R. H. Bruck has described a very general construction method which he called the extension of a set by a quasigroup. We use it to construct a class of examples for LF-quasigroups in which the image of the map $e(x) = x\backslash x$ is a group. More generally, we consider the variety of quasigroups which is defined by the property that the map $e$ is an endomorphism and its subvariety where the image of the map $e$ is a group. We characterize quasigroups belonging to these varieties using their Bruck decomposition with respect to the map $e$.

On the structure of left and right F-, SM-, and E-quasigroups

Fri, 06 Aug 2010 10:55 EDT

Victor Shcherbacov

Source: J. Gen. Lie Theory Appl., Volume 3, Number 3, 197--259.

Abstract:
It is proved that any left F-quasigroup is isomorphic to the direct product of a left F-quasigroup with a unique idempotent element and isotope of a special form of a left distributive quasigroup. The similar theorems are proved for right F-quasigroups, left and right SM- and E-quasigroups. Information on simple quasigroups from these quasigroup classes is given; for example, finite simple F-quasigroup is a simple group or a simple medial quasigroup. It is proved that any left F-quasigroup is isotopic to the direct product of a group and a left S-loop. Some properties of loop isotopes of F-quasigroups (including M-loops) are pointed out. A left special loop is an isotope of a left F-quasigroup if and only if this loop is isotopic to the direct product of a group and a left S-loop (this is an answer to Belousov ``1a'' problem). Any left E-quasigroup is isotopic to the direct product of an abelian group and a left S-loop (this is an answer to Kinyon-Phillips 2.8(1) problem). As corollary it is obtained that any left FESM-quasigroup is isotopic to the direct product of an abelian group and a left S-loop (this is an answer to Kinyon-Phillips 2.8(2) problem). New proofs of some known results on the structure of commutative Moufang loops are presented.

Symmetric bundles and representations of Lie triple systems

Fri, 06 Aug 2010 10:56 EDT

Wolfgang Bertram, Manon Didry

Source: J. Gen. Lie Theory Appl., Volume 3, Number 4, 261--284.

Abstract:
We define symmetric bundles as vector bundles in the category of symmetric spaces; it is shown that this notion is the geometric analog of the one of a representation of a Lie triple system. A symmetric bundle has an underlying reflection space, and we investigate the corresponding forgetful functor both from the point of view of differential geometry and from the point of view of representation theory. This functor is not injective, as is seen by constructing ``unusual'' symmetric bundle structures on the tangent bundles of certain symmetric spaces.

Unital algebras of Hom-associative type and surjective or injective twistings

Fri, 06 Aug 2010 10:56 EDT

Yaël Frégier, Aron Gohr, Sergei Silvestrov

Source: J. Gen. Lie Theory Appl., Volume 3, Number 4, 285--295.

Abstract:
In this paper, we introduce a common generalizing framework for alternative types of Hom-associative algebras. We show that the observation that unital Hom-associative algebras with surjective or injective twisting map are already associative has a generalization in this new framework. We also show by construction of a counterexample that another such generalization fails even in a very restricted particular case. Finally, we discuss an application of these observations by answering in the negative the question whether nonassociative algebras with unit such as the octonions may be twisted by the composition trick into Hom-associative algebras.

Arithmetic Witt-hom-Lie algebras

Fri, 06 Aug 2010 10:56 EDT

Daniel Larsson

Source: J. Gen. Lie Theory Appl., Volume 3, Number 4, 297--310.

Abstract:
This paper is concerned with explaining and further developing the rather technical definition of a hom-Lie algebra given in a previous paper which was an adaption of the ordinary definition to the language of number theory and arithmetic geometry. To do this we here introduce the notion of Witt-hom-Lie algebras and give interesting arithmetic applications, both in the Lie algebra case and in the hom-Lie algebra case. The paper ends with a discussion of a few possible applications of the developed hom-Lie language.

A connection whose curvature is the Lie bracket

Fri, 06 Aug 2010 10:56 EDT

Kent E. Morrison

Source: J. Gen. Lie Theory Appl., Volume 3, Number 4, 311--319.

Abstract:
Let $G$ be a Lie group with Lie algebra $\g$. On the trivial principal $G$-bundle over $\g$ there is a natural connection whose curvature is the Lie bracket of $\g$. The exponential map of $G$ is given by parallel transport of this connection. If $G$ is the diffeomorphism group of a manifold $M$, the curvature of the natural connection is the Lie bracket of vectorfields on $M$. In the case that $G=\SO(3)$ the motion of a sphere rolling on a plane is given by parallel transport of a pullback of the natural connection by a map from the plane to $\so(3)$. The motion of a sphere rolling on an oriented surface in $\R^3$ can be described by a similar connection.

On algebraic curves for commuting elements in q-Heisenberg algebras

Fri, 06 Aug 2010 10:56 EDT

Johan Richter, Sergei Silvestrov

Source: J. Gen. Lie Theory Appl., Volume 3, Number 4, 321--328.

Abstract:
In the present article we continue investigating the algebraic dependence of commuting elements in $q$-deformed Heisenberg algebras. We provide a simple proof that the $0$-chain subalgebra is a maximal commutative subalgebra when $q$ is of free type and that it coincides with the centralizer (commutant) of any one of its elements different from the scalar multiples of the unity. We review the Burchnall-Chaundy-type construction for proving algebraic dependence and obtaining corresponding algebraic curves for commuting elements in the $q$-deformed Heisenberg algebra by computing a certain determinant with entries depending on two commuting variables and one of the generators. The coefficients in front of the powers of the generator in the expansion of the determinant are polynomials in the two variables defining some algebraic curves and annihilating the two commuting elements. We show that for the elements from the $0$-chain subalgebra exactly one algebraic curve arises in the expansion of the determinant. Finally, we present several examples of computation of such algebraic curves and also make some observations on the properties of these curves.

Matrix Bosonic realizations of a Lie colour algebra with three generators and five relations of Heisenberg Lie type

Fri, 06 Aug 2010 10:56 EDT

Gunnar Sigurdsson, Sergei D. Silvestrov

Source: J. Gen. Lie Theory Appl., Volume 3, Number 4, 329--340.

Abstract:
We describe realizations of a Lie colour algebra with three generators and five relations by matrices of power series in noncommuting indeterminates satisfying Heisenberg's canonical commutation relation of quantum mechanics. The obtained formulas are used to construct new operator representations of this Lie colour algebra using canonical representation of the Heisenberg commutation relation and creation and annihilation operators of the quantum mechanical harmonic oscillator.

(α,β)-fuzzy Lie algebras over an (α,β)-fuzzy field

Fri, 06 Aug 2010 10:56 EDT

P. L. Antony, P. L. Lilly

Source: J. Gen. Lie Theory Appl., Volume 4, 8 pages.

Abstract:
The concept of (α,β)-fuzzy Lie algebras over an (α,β)-fuzzy field is introduced. We provide characterizations of an $(\in,\in\vee q)$-fuzzy Lie algebra over an $(\in,\in\vee q)$-fuzzy field.

Double cross biproduct and bi-cycle bicrossproduct Lie bialgebras

Fri, 06 Aug 2010 10:56 EDT

Tao Zhang

Source: J. Gen. Lie Theory Appl., Volume 4, 16 pages.

Abstract:
We construct double cross biproduct and bi-cycle bicrossproduct Lie bialgebras from braided Lie bialgebras. The main results generalize Majid's matched pair of Lie algebras and Drinfeld's quantum double and Masuoka's cross product Lie bialgebras.

Classifications of some classes of Zinbiel algebras

Fri, 06 Aug 2010 10:56 EDT

J. Q. Adashev, A. Kh. Khudoyberdiyev, B. A. Omirov

Source: J. Gen. Lie Theory Appl., Volume 4, 10 pages.

Abstract:
In this work, the nul-filiform and filiform Zinbiel algebras are described up to isomorphism. Moreover, the classification of complex Zinbiel algebras dimensions $\leq 3$ is extended up to dimension 4.

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

Thu, 29 Sep 2011 11:10 EDT

Elham Nobari, Mohammad Hosseini

Source: J. Gen. Lie Theory Appl., Volume 5, 12 pages.

Abstract:
Generalized polar decomposition method (or briefly GPD method) has been introduced by Munthe-Kaas and Zanna 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 $Z\in \mathfrak{g}$, a Lie algebra of matrices and computing $\exp(Z)\mathbf{v}$ for a vector $\mathbf{v}$. 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 $\mathbf{v}$.

Algebraic Structures Derived from Foams

Thu, 29 Sep 2011 11:10 EDT

J. Scott Carter, Masahico Saito

Source: J. Gen. Lie Theory Appl., Volume 5, 9 pages.

Abstract:
Foams are surfaces with branch lines at which three sheets merge. They have been used in the categorification of $\mathrm{sl}(3)$ 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.

An Approach to Omni-Lie Algebroids Using Quasi-Derivations

Thu, 29 Sep 2011 11:10 EDT

Dennise Garcia-Beltran, Jose A. Vallejo

Source: J. Gen. Lie Theory Appl., Volume 5, 10 pages.

Abstract:
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.

Some Properties of Intuitionistic Fuzzy Lie Algebras over a Fuzzy Field

Thu, 29 Sep 2011 11:10 EDT

P. L. Antony, P. L. Lilly

Source: J. Gen. Lie Theory Appl., Volume 5, 5 pages.

Abstract:
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.

On Graded Global Dimension of Color Hopf Algebras

Thu, 29 Sep 2011 11:10 EDT

Yan-Hua Wang

Source: J. Gen. Lie Theory Appl., Volume 5, 6 pages.

Abstract:
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

The Generalized Burnside Theorem in Noncommutative Deformation Theory

Thu, 29 Sep 2011 11:10 EDT

Eivind Eriksen

Source: J. Gen. Lie Theory Appl., Volume 5, 5 pages.

Abstract:
Let A be an associative algebra over a field $k$, and let $\mathcal{M}$ be a finite family of right $A$-modules. A study of the noncommutative deformation functor $\mathrm{Def}_{\mathcal{M}}$ of the family $\mathcal{M}$ leads to the construction of the algebra $\mathcal{O}^A(\mathcal{M})$ 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.

Meanders and Frobenius Seaweed Lie Algebras

Thu, 29 Sep 2011 11:10 EDT

Vincent Coll, Anthony Giaquinto, Colton Magnant

Source: J. Gen. Lie Theory Appl., Volume 5, 5 pages.

Abstract:
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.

Deformations of Complex 3-Dimensional Associative Algebras

Thu, 29 Sep 2011 11:10 EDT

Alice Fialowski, Michael Penkava, Mitch Phillipson

Source: J. Gen. Lie Theory Appl., Volume 5, 22 pages.

Abstract:
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.

Phase Spaces and Deformation Theory

Thu, 29 Sep 2011 11:10 EDT

Olav Arnfinn Laudal

Source: J. Gen. Lie Theory Appl., Volume 5, 18 pages.

Abstract:
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.

Geometry of Noncommutative k-Algebras

Thu, 29 Sep 2011 11:10 EDT

Arvid Siqveland

Source: J. Gen. Lie Theory Appl., Volume 5, 12 pages.

Abstract:
Let X be a scheme over an algebraically closed field k, and let $x\in\operatorname{Spec} R\subseteq X$ be a closed point corresponding to the maximal ideal $\mathfrak{m}\subseteq R$. Then $\hat{\mathcal{O}}_{X,x}$ is isomorphic to the prorepresenting hull, or local formal moduli, of the deformation functor $\mathrm{Def}_{R/\mathfrak{m}}:\underline{\ell}\rightarrow\mathrm{Sets}$. This suffices to reconstruct $X$ up to etalé 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.

Dynamical Yang-Baxter Maps Associated with Homogeneous Pre-Systems

Thu, 29 Sep 2011 11:10 EDT

Noriaki Kamiya, Youichi Shibukawa

Source: J. Gen. Lie Theory Appl., Volume 5, 9 pages.

Abstract:
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.