Necessary conditions for the existence of Hamiltonian actions of Lie groups and complexification of the induced Lie algebras of functions on the phase space. (A critical analysis)

1974 ◽  
Vol 10 (4) ◽  
pp. 149-155 ◽  
Author(s):  
F. Bunchaft
Keyword(s):  
Phase Space ◽  
Lie Groups ◽  
Lie Algebras ◽  
Critical Analysis ◽  
Necessary Conditions ◽  
Hamiltonian Actions

scholarly journals Invariant Poisson–Nijenhuis structures on Lie groups and classification

2018 ◽  
Vol 15 (04) ◽  
pp. 1850059 ◽  
Author(s):  
Zohreh Ravanpak ◽  
Adel Rezaei-Aghdam ◽  
Ghorbanali Haghighatdoost
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Symmetry Group ◽  
Lie Groups ◽  
Lie Algebras ◽  
Lie Group ◽  
Text Structure ◽  
Baxter Equation ◽  
Text Structures ◽  
Group A

We study right-invariant (respectively, left-invariant) Poisson–Nijenhuis structures ([Formula: see text]-[Formula: see text]) on a Lie group [Formula: see text] and introduce their infinitesimal counterpart, the so-called r-n structures on the corresponding Lie algebra [Formula: see text]. We show that [Formula: see text]-[Formula: see text] structures can be used to find compatible solutions of the classical Yang–Baxter equation (CYBE). Conversely, two compatible [Formula: see text]-matrices from which one is invertible determine an [Formula: see text]-[Formula: see text] structure. We classify, up to a natural equivalence, all [Formula: see text]-matrices and all [Formula: see text]-[Formula: see text] structures with invertible [Formula: see text] on four-dimensional symplectic real Lie algebras. The result is applied to show that a number of dynamical systems which can be constructed by [Formula: see text]-matrices on a phase space whose symmetry group is Lie group a [Formula: see text], can be specifically determined.

XIX.—Metrisable Lie Groups and Algebras

1957 ◽  
Vol 64 (3) ◽  
pp. 290-304 ◽  
Author(s):  
S.-T. Tsou ◽  
A. G. Walker
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Group ◽  
Necessary Conditions ◽  
Existence Theorems ◽  
Riemannian Metric ◽  
Abelian Algebra ◽  
Lie Groups And Algebras ◽  
Complex Extension

A Lie group is said to be metrisable if it admits a Riemannian metric which is invariant under all translations of the group. It is shown that the study of such groups reduces to the study of what are called metrisable Lie algebras, and some necessary conditions for a Lie algebra to be metrisable are given. Various decomposition and existence theorems are also given, and it is shown that every metrisable algebra is the product of an abelian algebra and a number of non-decomposable reduced algebras. The number of independent metrics admitted by a metrisable algebra is examined, and it is shown that the metric is unique when and only when the complex extension of the algebra is simple.

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

THE FERMI–WALKER DERIVATIVE IN LIE GROUPS

2013 ◽  
Vol 10 (07) ◽  
pp. 1320011 ◽  
Author(s):  
FATMA KARAKUŞ ◽  
YUSUF YAYLI
Keyword(s):  
Vector Field ◽  
Lie Groups ◽  
Lie Group ◽  
Necessary Conditions ◽  
Rotating Frame ◽  
Darboux Vector

In this study, Fermi–Walker derivative, Fermi–Walker parallelism, non-rotating frame, Fermi–Walker termed Darboux vector concepts are given for Lie groups in E4. First, we get any Frénet curve and any vector field along the Frénet curve in a Lie group. Then, the Fermi–Walker derivative is defined for the Lie group. Fermi–Walker derivative and Fermi–Walker parallelism are analyzed in Lie groups. Finally, the necessary conditions for Fermi–Walker parallelism are explained.

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.

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