Lie triple system

Nomizu's theorem relates invariant affine connections on reductive homogeneus spaces and nonassociative algebras. Among the algebras that appear in this relation we point out the so called Lie triple algebras which contain most of the infinitesimal information on reductive spaces. The aim of this work is to provide a brief survey on recent developements in this subject that show the interplay of nonassociative structures and connections and to pose an algebraic classification question on Lie triple algebras.

For finitely generated groups $G$ and $H$ equipped with word metrics, a translation-like action of $H$ on $G$ is a free action where each element of $H$ moves elements of $G$ a bounded distance. Translation-like actions provide a geometric generalization of subgroup containment. Extending work of Cohen, we show that cocompact lattices in a general semisimple Lie group $\mathbf{G}$ that is not isogenous to $\mathrm{SL}(2,\mathbb{R})$ admit translation-like actions by $\mathbb{Z}^2$. This result follows from a more general result. Namely, we prove that any cocompact lattice in the unipotent radical $\mathbf{N}$ of the Borel subgroup $\mathbf{AN}$ of $\mathbf{G}$ acts translation-like on any cocompact lattice in $\mathbf{G}$. We also prove that for noncompact simple Lie groups $G,H$ with $H<G$ and lattices $\Gamma < G$ and $\Delta < H$, that $\Gamma/\Delta$ is quasi-isometric to $G/H$ where $\Gamma/\Delta$ is the quotient via a translation-like action of $\Delta$ on $\Gamma$.

In this article, we introduce the notion of Lie triple centralizer as follows. Let A be an algebra, and φ : A → A be a linear mapping. We say that φ is a Lie triple centralizer whenever Then we characterize the general form of Lie triple centralizers on a generalized matrix algebra U and under some mild conditions on U we present the necessary and sufficient conditions for a Lie triple centralizer to be proper. As an application of our results, we characterize generalized Lie triple derivations on generalized matrix algebras.

Abstract. In this article, we introduce the notion of Lie triple centralizer as follows. Let A be an algebra, and φ : A → A be a linear mapping. We say that φ is a Lie triple centralizer whenever φ([[a, b], c]) = [[φ(a), b], c] for all a, b, c ∈ A. Then we characterize the general form of Lie triple centralizers on a generalized matrix algebra U and under some mild conditions on U we present the necessary and sufficient conditions for a Lie triple centralizer to be proper. As an application of our results, we characterize generalized Lie triple derivations on generalized matrix algebras.

This paper develops the structure theory of a Malcev algebra via the consideration of its most important and largest Lie (sub-) algebra. We introduce the notion of a Lie algebra which uniquely corresponds to a Malcev algebra and use this correspondence to derive some basic properties of some types of ideals in the Malcev algebra. We then prove the exact nature of the root-space decomposition of a Malcev algebra.

In his 2011 paper, Teleman proved that a cohomological field theory on the moduli space $\overline{\mathcal{M}}_{g,n}$ of stable complex curves is uniquely determined by its restriction to the smooth part $\mathcal{M}_{g,n}$, provided that the underlying Frobenius algebra is semisimple. This leads to a classification of all semisimple cohomological field theories. The present paper, the outcome of the author's master's thesis, presents Teleman's proof following his original paper. The author claims no originality: the main motivation has been to keep the exposition as complete and self-contained as possible.

Let R be an Artinian ring and let Γ E (R) be the compressed zero-divisor graph associated to R. The question of when the clique number ω(Γ E (R)) < ∞ was raised by J. Coykendall, S. Sather-Wagstaff, L. Sheppardson, and S. Spiroff, see [8, Section 5]. They proved that if ℓ(R) ≤ 4 then ω(Γ E (R)) < ∞. When ℓ(R) = 6, they gave an example of a local ring R where ω(Γ E (R)) = ∞ is possible by using the trivial extension of an Artinian local ring by its dualizing module. The question of what happens when ℓ(R) = 5 was stated as an open problem. We show that if ℓ(R) = 5 then ω(Γ E (R)) < ∞. We first reduce the problem to the case of a local ring (R, m, k). We then enumerate all possible Hilbert functions of R and show that the k-vector space m/m 2 admits a symmetric bilinear form in some cases of the Hilbert function. This allows us to relate the orthogonality in the bilinear space m/m 2 with the structure of zero-divisors in R. For instance, in the case when m 2 is principal and m 3 = 0, we show that R is Gorenstein if and only if the symmetric bilinear form on m/m 2 is non-degenerate. Moreover, in the case when ℓ(R) = 4, our techniques also yield a simpler and shorter proof of the finiteness of ω(Γ E (R)) avoiding, for instance, Cohen structure theorem.

In this PhD thesis, we consider two problems that are related to finite simple groups of Lie type. First of them is a problem mentioned in the Kourovka notebook: describe the finite simple groups in which every element is a product of two involutions. We consider the simple orthogonal groups in even characteristic, and solve the problem for them. Since other groups have been dealt with elsewhere, the problem is then solved completely. The second part of the thesis is related to Lie algebras. Every complex simple Lie algebra has a compact real form that is associated with a compact Lie group. In this thesis, we consider the Lie algebra of type E8, and give a new construction of its compact real form. The Lie product is defined using the irreducible subgroup of shape 25+10 ·GL5(2) of the automorphism group.

We apply Kolesnikov's algorithm to obtain a variety of nonassociative algebras defined by right anticommutativity and a "noncommutative" version of the Malcev identity. We use computational linear algebra to verify that these identities are equivalent to the identities of degree ≤ 4 satisfied by the dicommutator in every alternative dialgebra. We extend these computations to show that any special identity for Malcev dialgebras must have degree at least 7. Finally, we introduce a trilinear operation which makes any Malcev dialgebra into a Leibniz triple system.

Lie-Yamaguti algebras (or generalized Lie triple systems) are binary-ternary algebras intimately related to reductive homogeneous spaces. The Lie-Yamaguti algebras which are irreducible as modules over their Lie inner derivation algebra are the algebraic counterpart of the isotropy irreducible homogeneous spaces. These systems will be shown to split into three disjoint types: adjoint type, non-simple type and generic type. The systems of the first two types will be classified and most of them will be shown to be related to a Generalized Tits Construction of Lie algebras.

Lie-Yamaguti algebras (or generalized Lie triple systems) are intimately related to reductive homogeneous spaces. Simple Lie-Yamaguti algebras whose standard enveloping Lie algebra is the simple Lie algebra of type G2 are described, making use of the octonions. These examples reveal the much greater complexity of these systems, compared to Lie triple systems.

Lie triple system T over a field F of characteristic zero. It turns out that it contains nontrivial elements if and only if T is related to a simple Jordan algebra. In particular this provides a new proof of the determination by Laquer of the invariant affine connections in the simply connected compact irreducible Riemannian symmetric spaces.

Many geometric properties of a reductive homogeneous space are encoded in a binary and a ternary products defined on the tangent space at any point. This leads to the concept of a Lie-Yamaguti algebra. Some recent work on the relationships of the Lie-Yamaguti algebras associated to the isotropy irreducible homogeneous spaces and some other nonassociative systems are surveyed.

Lie-Yamaguti algebras (or generalized Lie triple systems) are binary-ternary algebras intimately related to reductive homogeneous spaces. The Lie-Yamaguti algebras which are irreducible as modules over their inner derivation algebras are the algebraic counterpart of the isotropy irreducible homogeneous spaces. These systems splits into three disjoint types: adjoint type, nonsimple type and generic type. The systems of the first two types were classified in a previous paper through a generalized Tits Construction of Lie algebras. In this paper, the Lie-Yamaguti algebras of generic type are classified by relating them to several other nonassociative algebraic systems: Lie and Jordan algebras and triple systems, Jordan pairs or Freudenthal triple systems.

We take an algorithmic and computational approach to a basic problem in abstract algebra: determining the correct generalization to dialgebras of a given variety of nonassociative algebras. We give a simplified statement of the KP algorithm introduced by Kolesnikov and Pozhidaev for extending polynomial identities for algebras to corresponding identities for dialgebras. We apply the KP algorithm to the defining identities for Jordan triple systems to obtain a new variety of nonassociative triple systems, called Jordan triple disystems. We give a generalized statement of the BSO algorithm introduced by Bremner and Sánchez-Ortega for extending multilinear operations in an associative algebra to corresponding operations in an associative dialgebra. We apply the BSO algorithm to the Jordan triple product and use computer algebra to verify that the polynomial identities satisfied by the resulting operations coincide with the results of the KP algorithm; this provides a large class of examples of Jordan triple disystems. We formulate a general conjecture expressed by a commutative diagram relating the output of the KP and BSO algorithms. We conclude by generalizing the Jordan triple product in a Jordan algebra to operations in a Jordan dialgebra; we use computer algebra to verify that resulting structures provide further examples of Jordan triple disystems. For this last result, we also provide an independent theoretical proof using Jordan structure theory.

1 Mathley là nhóm giải toán trên mạng xuất bản bài toán và lời giải định kỳ, bài viết phù hợp với học sinh trung học có năng khiếu toán học và các bạn trẻ yêu toán học, tham gia các cuộc thi học sinh giỏi toán. Mỗi năm có sáu ấn bản điện tử được ra đời nhằm phục vụ phong trào giải toán. Mathley is an online problem solving corner with prolems, solutions, and materials freely accessible to junior up to high school students. The corner is is made public six times per year on a regular basis dedicated to the promotion of problem solving among junior and high school students.

We take an algorithmic and computational approach to a basic problem in abstract algebra: determining the correct generalization to dialgebras of a given variety of nonassociative algebras. We give a simplified statement of the KP algorithm introduced by Kolesnikov and Pozhidaev for extending polynomial identities for algebras to corresponding identities for dialgebras. We apply the KP algorithm to the defining identities for Jordan triple systems to obtain a new variety of nonassociative triple systems, called Jordan triple disystems. We give a generalized statement of the BSO algorithm introduced by Bremner and Sánchez-Ortega for extending multilinear operations in an associative algebra to corresponding operations in an associative dialgebra. We apply the BSO algorithm to the Jordan triple product and use computer algebra to verify that the polynomial identities satisfied by the resulting operations coincide with the results of the KP algorithm; this provides a large class of examples of Jordan triple disystems. We formulate a general conjecture expressed by a commutative diagram relating the output of the KP and BSO algorithms. We conclude by generalizing the Jordan triple product in a Jordan algebra to operations in a Jordan dialgebra; we use computer algebra to verify that resulting structures provide further examples of Jordan triple disystems. For this last result, we also provide an independent theoretical proof using Jordan structure theory.

The paper deals with the Lie group algebraic structure of the set of Euclidean displacements, which represent rigid-body motions. We begin by looking for a representation of a displacement, which is independent of the choice of a frame of reference. Then, it is a simple matter to prove that displacement subgroups may be invariant by conjugation. This mathematical tool is suitable for solving special problems of mobility in mechanisms.

We consider simply connected, 2-step nilpotent Lie groups N, all of which are diffeomorphic to Euclidean spaces via the Lie group exponential map exp : ˆ→ N. We show that every such N with a suitable left invariant metric is the base space of a Riemannian submersion and homomorphism ρ : N* → N, where the fibers of ρ are flat, totally geodesic Euclidean spaces. The left invariant metric and Lie algebra of N* are obtained from N by constructing a Lie algebra G whose Killing form B is negative semidefinite. If B is negative definite, then we show that N* admits a (cocompact) lattice subgroup Γ*. Moreover, Γ = ρ(Γ*) is a lattice in N if Γ* ∩ Ker(ρ) is a lattice in Ker(ρ). Conversely, if N admits a lattice Γ, then N* admits a lattice Γ* such that Γ = ρ(Γ*). In this case the Riemannian submersion and homomorphism ρ : N* → N induces a Riemannian submersion ρ' : Γ* \ N* → Γ \ N whose fibers are flat, totally geodesic tori. The idea underlying the proof is that every 2-step nilpotent Lie algebra is isomorphic to a standard metric 2-step nilpotent Lie algebra, which we define and discuss. We also use a criterion of Mal'cev to derive conditions that guarantee the existence of lattices in N. We apply these conditions to prove the existence of lattices in simply connected, 2-step nilpotent Lie groups N that arise from Lie triple systems with compact center in so(n,Â), the Lie algebra of skew symmetric linear transformations of  n with the standard inner product. Lie triple systems with compact center include subspaces of so(n,Â) that arise from finite dimensional real representations of Clifford algebras or compact Lie groups. The center of the Lie triple system is trivial for representations of Clifford algebras and compact semisimple Lie groups.

The simple 7-dimensional Malcev algebra M is isomorphic to the irreducible sl(2, C)-module V (6) with binary product [x, y] = α(x ∧ y) defined by the sl(2, C)-module morphism α : Λ 2 V (6) → V (6). Combining this with the ternary product (x, y, z) = β(x∧y)•z defined by the sl(2, C)-module morphism β : Λ 2 V (6) → V (2) ≈ sl(2, C) gives M the structure of a generalized Lie triple system, or Lie-Yamaguti algebra. We use computer algebra to determine the polynomial identities of low degree satisfied by this binary-ternary structure.

Symmetric spaces are well known in differential geometry from the study of spaces of constant curvature. The tangent space of a symmetric space forms a Lie triple system. Recently these objects have received attention in the numerical analysis community. A remarkable number of different algorithms can be understood and analyzed using the concepts of symmetric spaces and this theory unifies a range of different topics in numerical analysis, such as polar type matrix decompositions, splitting methods for computation of the matrix exponential, composition of self adjoint numerical integrators and time symmetric dynamical systems. In this paper we will give an introduction to the mathematical theory behind these constructions, and review recent results. Furthermore, we are presenting new results related to time reversal symmetries, self adjoint numerical schemes and Yoshida type composition techniques.

Being a Lie group, the group SE(3) of orientation preserving motions of the real Euclidean 3–space becomes a symmetric space (in the sense of O. Loos) when endowed with the multiplication \mu(g, h) = gh^{−1}g. In this note we classify all connected symmetric subspaces of SE(3) up to conjugation. Moreover, we indicate some of its important applications in robot kinematics.

Just as the 3-D Euclidean space can be inverted through any of its points, the special Euclidean group SE(3) admits an inversion symmetry through any of its elements and is known to be a symmetric space. In this paper, we show that the symmetric submanifolds of SE(3) can be systematically exploited to study the kinematics of a variety of kinesiological and mechanical systems and, therefore, have many potential applications in robot kinematics. Unlike Lie subgroups of SE(3), symmetric submanifolds inherit distinct geometric properties from inversion symmetry. They can be generated by kinematic chains with symmetric joint twists. The main contribution of this paper is: 1) to give a complete classification of symmetric submanifolds of SE(3); 2) to investigate their geometric properties for robotics applications; and 3) to develop a generic method for synthesizing their kinematic chains.

The present work deals with problem of identification of isomorphism which is frequently encountered in structural synthesis of kinematic chains. Using the concept of weighed structural matrices some new codes has been proposed to reveal simultaneously link is isomorphic and chain is isomorphic. Firstly, two new codes for each link, called as extended adjacency link value (EALV) and link total loop size (TLS) are developed to detect distinct mechanisms of a kinematic chain. After that for each kinematic chain a new code called as extended adjacency string (EAS) which is a by-product of the same method is developed. The proposed method is easy to compute, reliable and capable of detecting isomorphism in all simple jointed compound kinematic chains. Some examples are also provided to demonstrate the effectiveness of this method. . (2013) 'Some new codes for isomorphism identification among kinematic chains and their inversions', Int. His research areas are kinematics of machines, computational mechanics and design optimisation. Ali Hasan has about 15 years teaching and five years industrial experience. Presently he is an Assistant Professor in the Mechanical Engineering Department, Faculty of Engineering and Technology, Jamia Millia Islamia University, New Delhi, India. His research areas are theory of machines, mechatronics and computational mechanics. He has published 17 research papers in national and international journals.