Lie Algebras Of Order Research Articles

Polynomial algebras from Lie algebra reduction chains [formula omitted] [formula omitted][formula omitted][formula omitted

We reexamine different examples of reduction chains g⊃g′ of Lie algebras in order to show how the polynomials determining the commutant with respect to the subalgebra g′ leads to polynomial deformations of Lie algebras. These polynomial algebras have already been observed in various contexts, such as in the framework of superintegrable systems. Two relevant chains extensively studied in Nuclear Physics, namely the Elliott chain su(3)⊃so(3) and the chain so(5)⊃su(2)×u(1) related to the Seniority model, are analysed in detail from this perspective. We show that these two chains both lead to three-generator cubic polynomial algebras, a result that paves the way for a more systematic investigation of nuclear models in relation to polynomial structures arising from reduction chains. In order to show that the procedure is not restricted to semisimple algebras, we also study the chain Sˆ(3)⊃sl(2,R)×so(2) involving the centrally-extended Schrödinger algebra in (3+1)-dimensional space–time. The approach chosen to construct these polynomial algebras is based on the use of the Lie-Poisson bracket, so that all the results are presented in the Poisson (commutative) setting. The advantage of considering this approach vs the enveloping algebras is emphasized, commenting on the main formal differences between the polynomial Poisson algebras and their noncommutative analogue. As an illustrative example of the latter, the three-generator cubic algebra associated to the Elliott chain is reformulated in the Lie algebraic (noncommutative) setting.

Rota–Baxter Operators of Nonzero Weight on a Complete Linear Lie Algebra of Order Two

Rota–Baxter Operators of Nonzero Weight on a Complete Linear Lie Algebra of Order Two

Several Isospectral and Non-Isospectral Integrable Hierarchies of Evolution Equations

By introducing a 3×3 matrix Lie algebra and employing the generalized Tu scheme, a AKNS isospectral–nonisospectral integrable hierarchy is generated by using a third-order matrix Lie algebra. Through a matrix transformation, we turn the 3×3 matrix Lie algebra into a 2×2 matrix case for which we conveniently enlarge it into two various expanding Lie algebras in order to obtain two different expanding integrable models of the isospectral–nonisospectral AKNS hierarchy by employing the integrable coupling theory. Specially, we propose a method for generating nonlinear integrable couplings for the first time, and produce a generalized KdV-Schrödinger integrable system and a nonlocal nonlinear Schrödinger equation, which indicates that we unite the KdV equation and the nonlinear Schrödinger equation as an integrable model by our method. This method presented in the paper could apply to investigate other integrable systems.

CCEPTED Automorphisms on complex simple Lie algebras of order 3

For complex simple Lie algebras, the article provides classification of all automorphisms of order 3. The method is an extension of Dynkin diagrams, so that the classification is a listing of diagrams which represent automorphisms of order 3. This work extends an earlier result on automorphisms of order 2. As an application, it shows that for automorphisms of orders 2 and 3 only, the invariant subalgebra determines the automorphism.

On the structure of graded Lie algebras of order 3

On the structure of graded Lie algebras of order 3

Split Lie algebras of order 3

We introduce the class of split Lie algebras of order 3 as the natural generalization of split Lie superalgebras and split Lie algebras. By means of connections of roots, we show that such a split Lie algebra of order 3 is of the form [Formula: see text] with [Formula: see text] a linear subspace of [Formula: see text] and any [Formula: see text] a well-described (split) ideal of [Formula: see text] satisfying [Formula: see text], with [Formula: see text], if [Formula: see text]. Additionally, under certain conditions, the (split) simplicity of the algebra is characterized in terms of the connections of nonzero roots, and a second Wedderburn type theorem for the class of split Lie algebras of order 3 (asserting that [Formula: see text] is the direct sum of the family of its (split) simple ideals) is stated.

Classification of filiform Lie algebras of order 3

Lie algebras of order 3 constitute a generalization of Lie algebras and superalgebras. Throughout this paper the classification problem of filiform Lie algebras of order 3 is considered and therefore this work is a continuation papers seen in the literature. We approach this classification by extending Vergne’s result for filiform Lie algebras and by considering algebras of order 3 of high nilindex. We find the expression of the law to which any elementary filiform Lie algebra of order 3 is isomorphic.

Infinitesimal deformations of filiform Lie algebras of order 3

The Lie algebras of order F have important applications for the fractional supersymmetry, and on the other hand the filiform Lie (super)algebras have very important properties into the Lie Theory. Thus, the aim of this work is to study filiform Lie algebras of order F which were introduced in Navarro (2014). In this work we obtain new families of filiform Lie algebras of order 3, in which the complexity of the problem rises considerably respecting to the cases considered in Navarro (2014).

Filiform Lie algebras of order 3

The aim of this work is to generalize a very important type of Lie algebras and superalgebras, i.e., filiform Lie (super)algebras, into the theory of Lie algebras of order F. Thus, the concept of filiform Lie algebras of order F is obtained. In particular, for F = 3 it has been proved that by using infinitesimal deformations of the associated model elementary Lie algebra it can be obtained families of filiform elementary lie algebras of order 3, analogously as that occurs into the theory of Lie algebras [M. Vergne, “Cohomologie des algèbres de Lie nilpotentes. Application à l’étude de la variété des algèbres de Lie nilpotentes,” Bull. Soc. Math. France 98, 81–116 (1970)]. Also we give the dimension, using an adaptation of the \documentclass[12pt]{minimal}\begin{document}$\mathfrak {sl}(2,\mathbb {C})$\end{document}sl(2,C)-module Method, and a basis of such infinitesimal deformations in some generic cases.

Classical R-matrix theory for bi-Hamiltonian field systems

This is a survey of the application of the classical R-matrix formalism to the construction of infinite-dimensional integrable Hamiltonian field systems. The main point is the study of bi-Hamiltonian structures. Appropriate constructions on Poisson, noncommutative and loop algebras as well as the central extension procedure are presented. The theory is developed for (1 + 1)- and (2 + 1)-dimensional field and lattice soliton systems as well as hydrodynamic systems. The formalism presented contains sufficiently many proofs and important details to make it self-contained and complete. The general theory is applied to several infinite-dimensional Lie algebras in order to construct both dispersionless and dispersive (soliton) integrable field systems.

Hopf algebras for ternary algebras

We construct a universal enveloping algebra associated with the ternary extension of Lie (super)algebras called Lie algebra of order three. A Poincaré–Birkhoff–Witt theorem is proven is this context. It this then shown that this universal enveloping algebra can be endowed with a structure of Hopf algebra. The study of the dual of the universal enveloping algebra enables to define the parameters of the transformation of a Lie algebra of order of 3. It turns out that these variables are the variables which generate the three-exterior algebra.

Some results on cubic and higher order extensions of the Poincaré algebra

In these lectures we study some possible higher order (of degree greater than two) extensions of the Poincaré algebra. We first give some general properties of Lie superalgebras with some emphasis on the supersymmetric extension of the Poincaré algebra or Supersymmetry. Some general features on the so-called Wess-Zumino model (the simplest field theory invariant under Supersymmetry) are then given. We further introduce an additional algebraic structure called Lie algebras of order F, which naturally comprise the concepts of ordinary Lie algebras and superalgebras. This structure enables us to define various non-trivial extensions of the Poincaré algebra. These extensions are studied more precisely in two different contexts. The first algebra we are considering is shown to be an (infinite dimensional) higher order extension of the Poincaré algebra in (1 + 2)-dimensions and turns out to induce a symmetry which connects relativistic anyons. The second extension we are studying is related to a specific finite dimensional Lie algebra of order three, which is a cubic extension of the Poincaré algebra in D-space-time dimensions. Invariant Lagrangians are constructed.

Color Lie algebras and Lie algebras of order F

The notion of color algebras is generalized to the class of F-ary algebras, and corresponding decoloration theorems are established. This is used to give a construction of colored structures by means of tensor products with Cli ord-like algebras. It is, moreover, shown that color algebras admit realizations as q = 0 quon algebras.

Nonstandard hulls of locally exponential Lie algebras

We show how to construct the nonstandard hull of certain infinite- dimensional Lie algebras in order to generalize a theorem of Pestov on the en- largeability of Banach-Lie algebras, yielding a partial answer to a question of Neeb from (11). In the process, we consider a nonstandard smoothness condition on functions between locally convex spaces to ensure that the induced function between the nonstandard hulls is smooth. We also discuss some conditions on a function between locally convex spaces which guarantee that its nonstandard extension maps finite points to finite points. 2000 Mathematics Subject Classification 22E65 (primary); 26E35, 26E20 (sec- ondary)

Color Lie algebras and Lie algebras of order F

The notion of color algebras is generalized to the class of F-ary algebras, and corresponding decoloration theorems are established. This is used to give a construction of colored structures by means of tensor products with Clifford-like algebras. It is moreover shown that color algebras admit realisations as q=0 quon algebras.

Ternary algebras and groups

We construct explicitly groups associated to specific ternary algebras which extend the Lie (super)algebras (called Lie algebras of order three). It turns out that the natural variables which appear in this construction are variables which generate the three-exterior algebra. An explicit matrix representation of a group associated to a peculiar Lie algebra of order three is constructed considering matrices with entry which belong to the three exterior algebra.

Kinematical superalgebras and Lie algebras of order 3

We study and classify kinematical algebras which appear in the framework of Lie superalgebras or Lie algebras of order 3. All these algebras are related through generalized Inonü–Wigner contractions from either the orthosymplectic superalgebra or the de Sitter Lie algebra of order 3.

Contractions and deformations of quasiclassical Lie algebras preserving a nondegenerate quadratic Casimir operator

By means of contractions of Lie algebras, we obtain new classes of indecomposable quasiclassical Lie algebras that satisfy the Yang-Baxter equations in its reformulation in terms of triple products. These algebras are shown to arise naturally from noncompact real simple algebras with nonsimple complexification, where we impose that a nondegenerate quadratic Casimir operator is preserved by the limiting process. We further consider the converse problem and obtain sufficient conditions on integrable cocycles of quasiclassical Lie algebras in order to preserve nondegenerate quadratic Casimir operators by the associated linear deformations.

A generalization of the Kantor-Koecher-Tits construction

The Kantor-Koecher-Tits construction associates a Lie algebra to any Jordan algebra. We generalize this construction to include also extensions of the associated Lie algebra. In particular, the conformal realization of so(p + 1, q + 1) generalizes to so(p + n, q + n), for arbitrary n, with a linearly realized subalgebra so(p, q). We also show that the construction applied to 3 ◊ 3 matrices over the division algebras R, C, H, O gives rise to the exceptional Lie algebras f4, e6, e7, e8, as well as to their ane, hyperbolic and further extensions. 2000 MSC: 17Cxx This article aims to give a brief overview of results that have already been shown by the author in [7], where details and a more comprehensive list of references are provided. We will mostly consider algebras over the real numbers, even though we will complexify real Lie algebras in order to study properties of the corresponding Dynkin diagrams. However, algebras are assumed to be real if nothing else is stated. A Jordan algebra is a commutative algebra that satisfies the Jordan identity a 2 ‐ (b ‐ a) = (a 2 ‐ b) ‐ a (1) The symmetric part of the product in an associative but noncommutative algebra, 2(a ‐ b) = (ab + ba)

Poincaré and sl(2) algebras of order 3

In this paper, we initiate a general classification for Lie algebras of order 3 and we give all Lie algebras of order 3 based on sl(2,C) and iso(1, 3) the Poincaré algebra in four dimensions. We then set the basis of the theory of the deformations (in the Gerstenhaber sense) and contractions for Lie algebras of order 3.