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

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.

Almost contact Hom-Lie algebras and Sasakian Hom-Lie algebras

2020 ◽  
pp. 2250005
Author(s):  
E. Peyghan ◽  
L. Nourmohammadifar
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Symmetric Matrix ◽  
Contact Structures ◽  
Sasakian Structure ◽  
Sasakian Structures ◽  
Matrix Formula ◽  
Skew Symmetric Matrix

In this paper, we consider Hom-Lie groups and introduce left invariant almost contact structures on them (almost contact Hom-Lie algebras). On such Hom-Lie groups, we construct the almost contact metrics and the contact forms. We give the notion of normal almost contact Hom-Lie algebras and describe [Formula: see text]-contact and Sasakian structures on Hom-Lie algebras. Also, we study some of their properties. In addition, it is shown that any Sasakian Hom-Lie algebra is a [Formula: see text]-contact Hom-Lie algebra. Finally, we present examples of Sasakian Hom-Lie algebras and in particular, we show that the skew symmetric matrix [Formula: see text] carries a Sasakian structure.

scholarly journals Parameter rigid actions of simply connected nilpotent Lie groups

2012 ◽  
Vol 33 (6) ◽  
pp. 1864-1875 ◽  
Author(s):  
HIROKAZU MARUHASHI
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Lie Group ◽  
Compact Manifold ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Heisenberg Groups ◽  
Dense Orbit ◽  
Simply Connected

AbstractWe show that for a locally free $C^{\infty }$-action of a connected and simply connected nilpotent Lie group on a compact manifold, if every real-valued cocycle is cohomologous to a constant cocycle, then the action is parameter rigid. The converse is true if the action has a dense orbit. Using this, we construct parameter rigid actions of simply connected nilpotent Lie groups whose Lie algebras admit rational structures with graduations. This generalizes the results of dos Santos [Parameter rigid actions of the Heisenberg groups. Ergod. Th. & Dynam. Sys.27(2007), 1719–1735] concerning the Heisenberg groups.

ON COHOMOLOGY OF NILPOTENT SYMPLECTIC LIE ALGEBRAS OF DIM. ≤ 6 AND DEFORMATION OF THOSE OF DIM. ≤ 4

2012 ◽  
Vol 09 (03) ◽  
pp. 1250020
Author(s):  
F. MOUNA ◽  
T. B. BOUETOU ◽  
M. B. NGUIFFO
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Contact Structures ◽  
Cohomology Groups ◽  
Nilpotent Lie Algebras ◽  
Differential Geom

This work is devoted to the result on symplectic nilpotent Lie algebras classified in a paper by [Y. Khakimdjanov, M. Goze and A. Medina, Symplectic or contact structures on Lie groups, Differential Geom. Appl. 21 (2004) 41–54]. We focus the attention on the investigation of their cohomology groups by using algebra tools. We give also an overview of deformation to those algebras of dim. ≤ 4.

scholarly journals A study on killing vector fields in four-dimensional spaces

Filomat ◽  
2019 ◽  
Vol 33 (4) ◽  
pp. 1249-1257
Author(s):  
Bahar Kırık
Keyword(s):  
Ricci Tensor ◽  
Vector Fields ◽  
Killing Vector ◽  
Positive Definite ◽  
Killing Vector Field ◽  
Killing Vector Fields ◽  
Metric Tensor ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Weyl Conformal Curvature Tensor

In the present study, some properties of Killing vector fields are investigated on 4-dimensional manifolds in case of the signature of the metric tensor 1 is either Lorentz or positive definite or neutral. First of all, the notation and the main object of the study are introduced on these manifolds. Later on, some special subalgebras are examined for the members of the Killing algebra when the Killing vector field vanishes at a point of the manifold admitting any of these metric signatures. The constraints of this examination to the Weyl conformal curvature tensor and the Ricci tensor are then studied and some results are obtained. Finally, some examples related to these results are given for all metric signatures.

STABLE RANK OF THE C*-ALGEBRAS OF NILPOTENT LIE GROUPS

1995 ◽  
Vol 06 (03) ◽  
pp. 439-446 ◽  
Author(s):  
TAKAHIRO SUDO ◽  
HIROSHI TAKAI
Keyword(s):  
Fixed Point ◽  
Lie Groups ◽  
Lie Algebras ◽  
Stable Rank ◽  
Nilpotent Lie Groups ◽  
The Real ◽  
C Algebras ◽  
Simply Connected

The stable rank of the C*-algebras of simply connected nilpotent Lie groups is determined by the dimension of the fixed point subspaces of the real adjoint spaces of the Lie algebras under the coadjoint actions, which is a generalization of the result obtained by A. J-L. Sheu. This equality is no longer affirmative in the case of non-nilpotent Lie groups.

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
Sign in / Sign up

Export Citation Format

Share Document