Lie algebras Research Papers - Academia.edu (original) (raw)

The class of minimal non-elementary Lie algebras over a field F are studied. These are classified when F is algebraically closed and of characteristic different from 2,3. The solvable algebras in this class are also characterised over any... more

The class of minimal non-elementary Lie algebras over a field F are studied. These are classified when F is algebraically closed and of characteristic different from 2,3. The solvable algebras in this class are also characterised over any perfect field.

In this article we develop an approach to deformations of the Witt and Virasoro algebras based on σ-derivations. We show that σ-twisted Jacobi type identity holds for generators of such deformations. For the σ-twisted generalization of... more

In this article we develop an approach to deformations of the Witt and Virasoro algebras based on σ-derivations. We show that σ-twisted Jacobi type identity holds for generators of such deformations. For the σ-twisted generalization of Lie algebras modeled by this construction, we develop a theory of central extensions. We show that our approach can be used to construct new deformations of Lie algebras and their central extensions, which in particular include naturally the q-deformations of the Witt and Virasoro algebras associated to q-difference operators, providing also corresponding q-deformed Jacobi identities.

Given an n-dimensional Lie algebra gg over a field k⊃Qk⊃Q, together with its vector space basis X10,…,Xn0, we give a formula, depending only on the structure constants, representing the infinitesimal generators, Xi=Xi0t in... more

Given an n-dimensional Lie algebra gg over a field k⊃Qk⊃Q, together with its vector space basis X10,…,Xn0, we give a formula, depending only on the structure constants, representing the infinitesimal generators, Xi=Xi0t in gk⊗k[[t]]g⊗kk[[t]], where t is a formal variable, as a formal power series in t with coefficients in the Weyl algebra AnAn. Actually, the theorem is proved for Lie algebras over arbitrary rings k⊃Qk⊃Q.We provide three different proofs, each of which is expected to be useful for generalizations. The first proof is obtained by direct calculations with tensors. This involves a number of interesting combinatorial formulas in structure constants. The final step in calculation is a new formula involving Bernoulli numbers and arbitrary derivatives of coth(x/2)coth(x/2). The dimensions of certain spaces of tensors are also calculated. The second method of proof is geometric and reduces to a calculation of formal right-invariant vector fields in specific coordinates, in a (new) variant of formal group scheme theory. The third proof uses coderivations and Hopf algebras.

This paper considers Platonic solids/polytopes in the real Euclidean space R(n) of dimension 3 ≤ n < ∞. The Platonic solids/polytopes are described together with their faces of dimensions 0 ≤ d ≤ n - 1. Dual pairs of Platonic polytopes... more

This paper considers Platonic solids/polytopes in the real Euclidean space R(n) of dimension 3 ≤ n < ∞. The Platonic solids/polytopes are described together with their faces of dimensions 0 ≤ d ≤ n - 1. Dual pairs of Platonic polytopes are considered in parallel. The underlying finite Coxeter groups are those of simple Lie algebras of types A(n), B(n), C(n), F4, also called the Weyl groups or, equivalently, crystallographic Coxeter groups, and of non-crystallographic Coxeter groups H3, H4. The method consists of recursively decorating the appropriate Coxeter-Dynkin diagram. Each recursion step provides the essential information about faces of a specific dimension. If, at each recursion step, all of the faces are in the same Coxeter group orbit, i.e. are identical, the solid is called Platonic. The main result of the paper is found in Theorem 2.1 and Propositions 3.1 and 3.2.

In this paper, we generalize the geometry of the product pseudo-Riemannian manifold equipped with the product Poisson structure ([10]) to the geometry of a warped product of pseudo-Riemannian manifolds equipped with a warped Poisson... more

In this paper, we generalize the geometry of the product pseudo-Riemannian manifold equipped with the product Poisson structure ([10]) to the geometry of a warped product of pseudo-Riemannian manifolds equipped with a warped Poisson structure. We construct three bivector fields on a product manifold and show that each of them lead under certain conditions to a Pois-son structure. One of these bivector fields will be called the warped bivector field. For a warped product of pseudo-Riemannian manifolds equipped with a warped bivector field, we compute the corresponding contravariant Levi-Civita connection and the curvatures associated with.

For two different natural definitions of Casimir operators for simple Lie algebras we show that their eigenvalues in the adjoint representation can be expressed polynomially in the universal Vogel's parameters alpha,beta,gamma\alpha, \beta, \gammaalpha,beta,gamma and... more

For two different natural definitions of Casimir operators for simple Lie algebras we show that their eigenvalues in the adjoint representation can be expressed polynomially in the universal Vogel's parameters alpha,beta,gamma\alpha, \beta, \gammaalpha,beta,gamma and give explicit formulae for the generating functions of these eigenvalues.

We present the derivation of the 6-dimensional Eulerian Lie group of the form SO(3,C). We describe our derivation process, which involves the creation of a finite group by using permutation matrices, and the exponentiation of the adjoint... more

We present the derivation of the 6-dimensional Eulerian Lie group of the form SO(3,C). We describe our derivation process, which involves the creation of a finite group by using permutation matrices, and the exponentiation of the adjoint form of the subset representing the generators of the finite group. We take clues from the 2-dimensional complex rotation matrix to present, what we believe, is a true representation of the Lie group for the six-dimensional complex unit sphere and proceed to study its dynamics. With this approach, we can proceed to present this SO(3,C) group and derive its unitary counterpart that is U(3). The following findings can prove useful in mathematical physics, complex analysis and applications in deriving higher dimensional forms of similar division algebras.

In this paper, using the vector and tensor calculus and the multidual algebra, a new computing method for studying the higher-order acceleration field properties in the generally rigid body motion is proposed. The results are... more

In this paper, using the vector and tensor calculus and the multidual algebra, a new computing method for studying the higher-order acceleration field properties in the generally rigid body motion is proposed. The results are coordinate-free and in a closed form. Higher-order kinematics analysis of lower-pair serial chains with multidual
algebra will be done. The velocity, acceleration, jerk, and jounce fields are given. This approach uses the morphism between the Lie group of the rigid displacements and the Lie group of orthogonal multidual tensors.

We explore the graded and filtered formality properties of a finitely-generated group by studying the various Lie algebras attached to such a group, including the associated graded Lie algebra, the holonomy Lie algebra, and the Malcev Lie... more

We explore the graded and filtered formality properties of a finitely-generated group by studying the various Lie algebras attached to such a group, including the associated graded Lie algebra, the holonomy Lie algebra, and the Malcev Lie algebra. We explain how these notions behave with respect to split injections, coproducts, and direct products, and how they are inherited by solvable and nilpotent quotients.
For a finitely-presented group, we give an explicit formula for the cup product in low degrees, and an algorithm for computing the holonomy Lie algebra, using a Magnus expansion method. We also give a presentation for the Chen Lie algebra of a filtered-formal group, and discuss various approaches to computing the ranks of the graded objects under consideration. We illustrate our approach with examples drawn from a variety of group-theoretic and topological contexts, such as 1-relator groups, finitely generated torsion-free nilpotent groups, link groups, and fundamental groups of Seifert fibered manifolds.

Amongst other results, we perform a ‘contactization’ method to construct, in every odd dimension, many contact Lie groups with a discrete center, unlike the usual (classical) contactization which only produces Lie groups with a... more

Amongst other results, we perform a ‘contactization’ method to construct, in every odd dimension, many contact Lie groups with a discrete center, unlike the usual (classical) contactization which only produces Lie groups with a non-discrete center. We discuss some applications and consequences of such a construction, construct several examples and derive some properties. We give classification results in low dimensions. A complete list is supplied in dimension 5. In any odd dimension greater than 5, there are infinitely many locally non-isomorphic solvable contact Lie groups. We also characterize solvable contact Lie algebras whose derived ideal has codimension one. For simplicity, most of the results are given in the Lie algebra version.

The aim of this paper is to prove that there is an isomorphism of graded A-modules and of chain complexes between Vn(g), the Chevalley-Eilenberg complex for a Lie algebra g and Wn(g), the complex formed from of Dual numbers. The paper... more

The aim of this paper is to prove that there is an isomorphism of graded A-modules and of chain complexes between Vn(g), the Chevalley-Eilenberg complex for a Lie algebra g and Wn(g), the complex formed from of Dual numbers.
The paper is divided in two parts; in the rst part it is built Wn(g), for it, the steps necessary to reach the desired complex are described. Subsequently, the maps that allow to establish the isomorphism between the two structures are de ned. This process will allow to establish an alternative description of the Chevalley-Eilenberg complex.

The present research is focused on developing new methods for recovering a solution to the Kepler full-body problem in a non-inertial reference frame. A representation theorem is provided for the full-body initial value problem, using... more

The present research is focused on developing new methods for recovering a solution to the Kepler full-body problem in a non-inertial reference frame. A representation theorem is provided for the full-body initial value problem, using dual Lie algebra approach and dual quaternions. Furthermore, the representation theorems for the rotation part and translation part of the Keplerianfull-body motion in a non-inertial reference frame are obtained. Regarding the translation part of the center of mass, a closed-form coordinate-free solution is revealed based on generalized trigonometric function in space at constant curvature. They hold for all types of inertial trajectories (elliptic, parabolic, hyperbolic, rectilinear). The proof of concept is sustained by computational solutions.

Following Lazard, we study the N-series of a group G and their associated graded Lie algebras. The main examples we consider are the lower central series and Stallings' rational and mod-p versions of this series. Building on the work of... more

Following Lazard, we study the N-series of a group G and their associated graded Lie algebras. The main examples we consider are the lower central series and Stallings' rational and mod-p versions of this series. Building on the work of Massuyeau and Guaschi-Pereiro, we describe these N-series and Lie algebras in the case when G splits as a semidirect product, in terms of the relevant data for the factors and the monodromy action. As applications, we recover a well-known theorem of Falk-Randell regarding split extensions with trivial monodromy on abelianization and its mod-p version due to Bellingeri-Gervais and prove an analogous result for the rational lower central series of split extensions with trivial monodromy on torsion-free abelianization.

We explore finitely generated groups by studying the nilpotent towers and the various Lie algebras attached to such groups. Our main goal is to relate an isomor-phism extension problem in the Postnikov tower to the existence of certain... more

We explore finitely generated groups by studying the nilpotent towers and the various Lie algebras attached to such groups. Our main goal is to relate an isomor-phism extension problem in the Postnikov tower to the existence of certain commuting diagrams. This recasts a result of G. Rybnikov in a more general framework and leads to an application to hyperplane arrangements, whereby we show that all the nilpotent quotients of a decomposable arrangement group are combinatorially determined.

We describe an object oriented MATLAB toolbox for solving di erential equations on manifolds. The software re ects recent development within the area of geometric integration. Through the use of elements from di erential geometry, in... more

We describe an object oriented MATLAB toolbox for solving di erential equations on manifolds. The software re ects recent development within the area of geometric integration. Through the use of elements from di erential geometry, in particular Lie groups and homogeneous spaces, coordinate free formulations of numerical integrators are developed. The strict mathematical de nitions and results are well suited for implementation in an object oriented language, and, due to its simplicity, the authors have chosen MATLAB as the working environment. The basic ideas of Di Man are presented, along with particular examples that illustrate the working of and the theory behind the software package. AMS Subject Classi cation: 65L06, 34A50

We study the integral, rational, and modular Alexander invariants, as well as the cohomology jump loci of groups arising as extensions with trivial algebraic monodromy. Our focus is on extensions of the form 1 -> K -> G -> Q -> 1, where Q... more

We study the integral, rational, and modular Alexander invariants, as well as the cohomology jump loci of groups arising as extensions with trivial algebraic monodromy. Our focus is on extensions of the form 1 -> K -> G -> Q -> 1, where Q is an abelian group acting trivially on H _1 (K; Z) with suitable modifications in the rational and mod-p settings. We find a tight relationship between the Alexander invariants, the characteristic varieties, and the resonance varieties of the groups K and G. This leads to an inequality between the respective Chen ranks, which becomes an equality in degrees greater than 1 for split extensions.

A finite simplicial graph Γ determines a right-angled Artin group G_Γ, with generators corresponding to the vertices of Γ, and with a relation υw=wυ for each pair of adjacent vertices. We compute the lower central series quotients, the... more

A finite simplicial graph Γ determines a right-angled Artin group G_Γ, with generators corresponding to the vertices of Γ, and with a relation υw=wυ for each pair of adjacent vertices. We compute the lower central series quotients, the Chen quotients, and the (first) resonance variety of G_Γ, directly from the graph Γ.

Let g be an untwisted affine Kac-Moody algebra and M_J(lambda) a Verma-type module for g with J-highest integral weight lambda. We construct quantum Verma-type modules M_J^q(lambda) over the quantum group U_q(g), investigate their... more

Let g be an untwisted affine Kac-Moody algebra and M_J(lambda) a Verma-type module for g with J-highest integral weight lambda. We construct quantum Verma-type modules M_J^q(lambda) over the quantum group U_q(g), investigate their properties and show that M_J^q(lambda) is a true quantum deformation of M_J(\l) in the sense that the weight structure is preserved under the deformation. We also analyze the submodule structure of quantum Verma-type modules.

Let g be an untwisted affine Kac-Moody algebra and M_J(lambda) a Verma-type module for g with J-highest integral weight lambda. We construct quantum Verma-type modules M_J^q(lambda) over the quantum group U_q(g), investigate their... more

Let g be an untwisted affine Kac-Moody algebra and M_J(lambda) a Verma-type module for g with J-highest integral weight lambda. We construct quantum Verma-type modules M_J^q(lambda) over the quantum group U_q(g), investigate their properties and show that M_J^q(lambda) is a true quantum deformation of M_J(\l) in the sense that the weight structure is preserved under the deformation. We also analyze the submodule structure of quantum Verma-type modules.

Abstract: The main goal of this paper is to present a unifying theory to describe the pure rolling motions of Riemannian symmetric spaces. We make a clear con-nection between the structure of the kinematic equations of rolling and the... more

Abstract: The main goal of this paper is to present a unifying theory to describe the pure rolling motions of Riemannian symmetric spaces. We make a clear con-nection between the structure of the kinematic equations of rolling and the natural decomposition of the Lie algebra associated to the symmetric space. This empha-sizes the relevance of Lie theory in the geometry of rolling manifolds. It becomes clear why many particular examples scattered through the existing literature always show a common pattern. 1.

For a finitely-presented group, we give an explicit formula for the cup product in low degrees, and an algorithm for computing the holonomy Lie algebra, using a Magnus expansion method. We also give a presentation for the Chen Lie algebra... more

For a finitely-presented group, we give an explicit formula for the cup product in low degrees, and an algorithm for computing the holonomy Lie algebra, using a Magnus expansion method. We also give a presentation for the Chen Lie algebra of a filtered-formal group, and discuss various approaches to computing the ranks of the graded objects under consideration. We illustrate our approach with examples drawn from a variety of group-theoretic and topological contexts, such as 1-relator groups, finitely generated torsion-free nilpotent groups, link groups, and fundamental groups of Seifert fibered manifolds.