scholarly journals ON LIE ALGEBRAS OF INFINITESIMAL AFFINE TRANSFORMATIONS IN TANGENT BUNDLES WITH A COMPLETE LIFT CONNECTION

2016 ◽  
Author(s):  
G. A. Sultanova ◽  
Keyword(s):  
Lie Algebras ◽  
Affine Transformations ◽  
Complete Lift ◽  
Tangent Bundles

scholarly journals Simply transitive NIL-affine actions of solvable Lie groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jonas Deré ◽  
Marcos Origlia
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Group ◽  
Main Tool ◽  
Full Description ◽  
Nilpotent Lie Group ◽  
Affine Transformations ◽  
Solvable Lie Algebra ◽  
Solvable Lie Group

Abstract Every simply connected and connected solvable Lie group 𝐺 admits a simply transitive action on a nilpotent Lie group 𝐻 via affine transformations. Although the existence is guaranteed, not much is known about which Lie groups 𝐺 can act simply transitively on which Lie groups 𝐻. So far, the focus was mainly on the case where 𝐺 is also nilpotent, leading to a characterization depending only on the corresponding Lie algebras and related to the notion of post-Lie algebra structures. This paper studies two different aspects of this problem. First, we give a method to check whether a given action ρ : G → Aff ⁡ ( H ) \rho\colon G\to\operatorname{Aff}(H) is simply transitive by looking only at the induced morphism φ : g → aff ⁡ ( h ) \varphi\colon\mathfrak{g}\to\operatorname{aff}(\mathfrak{h}) between the corresponding Lie algebras. Secondly, we show how to check whether a given solvable Lie group 𝐺 acts simply transitively on a given nilpotent Lie group 𝐻, again by studying properties of the corresponding Lie algebras. The main tool for both methods is the semisimple splitting of a solvable Lie algebra and its relation to the algebraic hull, which we also define on the level of Lie algebras. As an application, we give a full description of the possibilities for simply transitive actions up to dimension 4.

scholarly journals LIFTS OF GOLDEN STRUCTURES ON THE TANGENT BUNDLE

2021 ◽  
pp. 643
Author(s):  
Geeta Verma
Keyword(s):  
Tangent Bundle ◽  
Third Order ◽  
Integrability Conditions ◽  
Complete Lift ◽  
Tangent Bundles ◽  
Order Tangent

The present paper aims to study the complete lift of golden structure on tangent bundles. Integrability conditions for complete lift and third-order tangent bundle are established.

Infinitesimal affine transformations of a Weil bundle of second order with complete lift connection

2015 ◽  
Vol 59 (12) ◽  
pp. 1-9
Author(s):  
K. M. Budanov ◽  
A. Ya. Sultanov
Keyword(s):  
Second Order ◽  
Affine Transformations ◽  
Weil Bundle ◽  
Complete Lift

scholarly journals Infinitesimal affine transformations of the tangent bundles with Sasaki metric

1974 ◽  
Vol 26 (3) ◽  
pp. 353-361 ◽  
Author(s):  
Kōjirō Satō
Keyword(s):  
Affine Transformations ◽  
Sasaki Metric ◽  
Tangent Bundles

Lie Groups, Lie Algebras, Cohomology and some Applications in Physics

1995 ◽  
Author(s):  
Josi A. de Azcárraga ◽  
Josi M. Izquierdo
Keyword(s):  
Lie Groups ◽  
Lie Algebras

The Standard Metric

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Affine Transformation ◽  
Convex Sets ◽  
Finite Dimension ◽  
Affine Subspace ◽  
Affine Transformations ◽  
Real Vector ◽  
Euclidean Metric ◽  
Affine Hyperplane ◽  
Tits Building ◽  
Convex Closure

This chapter presents some results about groups generated by reflections and the standard metric on a Bruhat-Tits building. It begins with definitions relating to an affine subspace, an affine hyperplane, an affine span, an affine map, and an affine transformation. It then considers a notation stating that the convex closure of a subset a of X is the intersection of all convex sets containing a and another notation that denotes by AGL(X) the group of all affine transformations of X and by Trans(X) the set of all translations of X. It also describes Euclidean spaces and assumes that the real vector space X is of finite dimension n and that d is a Euclidean metric on X. Finally, it discusses Euclidean representations and the standard metric.

The rigidity of some solvable Lie algebras

2018 ◽  
Vol 2018 (2) ◽  
pp. 43-49
Author(s):  
R.K. Gaybullaev ◽  
Kh.A. Khalkulova ◽  
J.Q. Adashev
Keyword(s):  
Lie Algebras ◽  
Solvable Lie Algebras
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