Positive definite Minkowski Lie algebras and bi-invariant Finsler metrics on Lie groups

2008 ◽  
Vol 136 (1) ◽  
pp. 191-201 ◽  
Author(s):  
Shaoqiang Deng ◽  
Zixin Hou
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Positive Definite ◽  
Finsler Metrics

scholarly journals INVARIANT PSEUDO-SASAKIAN AND K-CONTACT STRUCTURES ON SEVEN-DIMENSIONAL NILPOTENT LIE GROUPS

2017 ◽  
pp. 91-99
Author(s):  
Nikolai Smolentsev ◽  
Nikolai Smolentsev
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Ricci Tensor ◽  
Positive Definite ◽  
Contact Structures ◽  
Nilpotent Lie Groups ◽  
Central Extensions ◽  
Metric Tensor ◽  
Group A ◽  
Contact Distribution

This paper studies the existence of left-invariant Sasaki contact structures on the seven-dimensional nilpotent Lie groups. It is shown that the only Lie group allowing Sasaki structure with a positive definite metric tensor is the Heisenberg group A complete list of 22 classes of seven-dimensional nilpotent Lie groups which admit pseudo-Riemannian Sasaki structures is found. A list of 25 classes of seven-dimensional nilpotent Lie groups admitting K-contact structures, but not pseudo-Riemannian Sasaki structures, is also presented. All the contact structures considered are central extensions of six-dimensional nilpotent symplectic Lie groups. Formulas that connect the geometric characteristics of six-dimensional nilpotent almost pseudo-Kähler Lie groups and seven-dimensional nilpotent contact Lie groups are established. As is known, for six-dimensional nilpotent pseudo-Kähler Lie groups the Ricci tensor is always zero. In contrast to the pseudo-Kӓhler case, it is shown that on contact seven-dimensional Lie algebras the Ricci tensor is nonzero even in directions of the contact distribution

Inverse problem of left invariant sprays on Lie groups

2021 ◽  
pp. 2150076
Author(s):  
Libing Huang ◽  
Xiaohuan Mo
Keyword(s):  
Inverse Problem ◽  
Lie Groups ◽  
Lie Group ◽  
Positive Definite ◽  
Finsler Metric ◽  
Finsler Metrics ◽  
Riemann Curvature

In this paper, we discuss inverse problem in spray geometry. We find infinitely many sprays with non-diagonalizable Riemann curvature on a Lie group, these sprays are not induced by Finsler metrics. We also study left invariant sprays with non-vanishing spray vectors on Lie groups. We prove that if such a spray [Formula: see text] on a Lie group [Formula: see text] satisfies that [Formula: see text] is commutative or [Formula: see text] is projective, then [Formula: see text] is not induced by any (not necessary positive definite) left invariant Finsler metric. Finally, we construct an abundance of the left invariant sprays on Lie groups which satisfy the conditions in above result.

scholarly journals Representations of complex semisimple Lie groups and Lie algebras

1966 ◽  
Vol 72 (3) ◽  
pp. 522-526 ◽  
Author(s):  
K. R. Parthasarathy ◽  
R. Ranga Rao ◽  
V. S. Varadarajan
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Semisimple Lie Groups ◽  
Complex Semisimple

Harmonicity of vector fields on a class of Lorentzian solvable Lie groups

2018 ◽  
Vol 18 (3) ◽  
pp. 337-344 ◽  
Author(s):  
Ju Tan ◽  
Shaoqiang Deng
Keyword(s):  
Vector Field ◽  
Lie Groups ◽  
Lie Algebras ◽  
Critical Points ◽  
Vector Fields ◽  
Energy Functional ◽  
Smooth Vector ◽  
Solvable Lie Groups ◽  
Invariant Vector ◽  
Invariant Vector Fields

AbstractIn this paper, we consider a special class of solvable Lie groups such that for any x, y in their Lie algebras, [x, y] is a linear combination of x and y. We investigate the harmonicity properties of invariant vector fields of this kind of Lorentzian Lie groups. It is shown that any invariant unit time-like vector field is spatially harmonic. Moreover, we determine all vector fields which are critical points of the energy functional restricted to the space of smooth vector fields.

scholarly journals Dual Lie algebras of Heisenberg Poisson Lie groups

1993 ◽  
Vol 17 (2) ◽  
pp. 429-441
Author(s):  
Kentaro Mikami ◽  
Fumio Narita
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Poisson Lie Groups
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