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.

scholarly journals Some examples of quasiisometries of nilpotent Lie groups

2016 ◽  
Vol 2016 (718) ◽  
Author(s):  
Xiangdong Xie
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Simply Connected

AbstractWe construct quasiisometries of nilpotent Lie groups. In particular, for any simply connected nilpotent Lie group

scholarly journals VARIANTS OF MIYACHI’S THEOREM FOR NILPOTENT LIE GROUPS

2010 ◽  
Vol 88 (1) ◽  
pp. 1-17 ◽  
Author(s):  
ALI BAKLOUTI ◽  
SUNDARAM THANGAVELU
Keyword(s):  
Fourier Transform ◽  
Lie Groups ◽  
Lie Group ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Group Fourier Transform ◽  
Simply Connected ◽  
Miyachi’S Theorem

AbstractWe formulate and prove two versions of Miyachi’s theorem for connected, simply connected nilpotent Lie groups. This allows us to prove the sharpness of the constant 1/4 in the theorems of Hardy and of Cowling and Price for any nilpotent Lie group. These theorems are proved using a variant of Miyachi’s theorem for the group Fourier transform.

scholarly journals On the isomorphism problem for C∗-algebras of nilpotent Lie groups

2019 ◽  
pp. 1-30
Author(s):  
Ingrid Beltiţă ◽  
Daniel Beltiţă
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Isomorphism Problem ◽  
Nilpotent Lie Groups ◽  
Direct Products ◽  
Nilpotent Lie Group ◽  
Heisenberg Groups ◽  
C Algebras ◽  
Dimension Formula ◽  
Unitary Dual

We investigate to what extent a nilpotent Lie group is determined by its [Formula: see text]-algebra. We prove that, within the class of exponential Lie groups, direct products of Heisenberg groups with abelian Lie groups are uniquely determined even by their unitary dual, while nilpotent Lie groups of dimension [Formula: see text] are uniquely determined by the Morita equivalence class of their [Formula: see text]-algebras. We also find that this last property is shared by the filiform Lie groups and by the [Formula: see text]-dimensional free two-step nilpotent Lie group.

scholarly journals Proper Actions of Certain Nilpotent Affine Groups with Codimension One Orbits

2012 ◽  
Vol 4 (2) ◽  
pp. 315
Author(s):  
N. Salma
Keyword(s):  
Homogeneous Space ◽  
Lie Groups ◽  
Lie Group ◽  
Nilpotent Groups ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Proper Actions ◽  
Codimension One ◽  
Affine Groups ◽  
Simply Connected

Criterion for proper actions has been established for a homogeneous space of reductive type by Kobayashi (Math. Ann. 1989, 1996). On the other hand, an analogous criterion to Kobayashi’s equivalent conditions was proposed by Lipsman (1995) for a nilpotent Lie group . Lipsman's Conjecture: Let  be a simply connected nilpotent Lie group. Then the following two conditions on connected subgroups  and  are equivalent: (i) the action of  on  is proper; (ii)  is compact for any  The condition (i) is important in the study of discontinuous groups for the homogeneous space , while the second condition (ii) can easily be checked. The implication (i)  (ii) is obvious, and the opposite implication (ii)  (i) was known only in some lower dimensional cases. In this paper we prove the equivalence (i) ? (ii) for certain affine nilpotent Lie groups . Keywords: Affine nilpotent groups; Homogeneous manifolds; Proper actions; Properly discontinuous actions; Simply connected nilpotent Lie groups; Compact isotropy property (CI);  Eigenvalues. © 2012 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi: http://dx.doi.org/10.3329/jsr.v4i2.7889 J. Sci. Res. 4 (2), 315-326 (2012)

CRITERION OF PROPER ACTIONS FOR 3-STEP NILPOTENT LIE GROUPS

2007 ◽  
Vol 18 (07) ◽  
pp. 783-795 ◽  
Author(s):  
TARO YOSHINO
Keyword(s):  
Homogeneous Space ◽  
Lie Groups ◽  
Lie Group ◽  
Closed Subgroup ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Proper Actions ◽  
Affirmative Solution

For a nilpotent Lie group G and its closed subgroup L, Lipsman [13] conjectured that the L-action on some homogeneous space of G is proper in the sense of Palais if and only if the action is free. Nasrin [14] proved this conjecture assuming that G is a 2-step nilpotent Lie group. However, Lipsman's conjecture fails for the 4-step nilpotent case. This paper gives an affirmative solution to Lipsman's conjecture for the 3-step nilpotent case.

scholarly journals Ricci-flat and Einstein pseudoriemannian nilmanifolds

2019 ◽  
Vol 6 (1) ◽  
pp. 170-193 ◽  
Author(s):  
Diego Conti ◽  
Federico A. Rossi
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Einstein Metrics ◽  
Einstein Metric ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Ricci Flat ◽  
Open Questions ◽  
Expository Paper ◽  
New Criterion

AbstractThis is partly an expository paper, where the authors’ work on pseudoriemannian Einstein metrics on nilpotent Lie groups is reviewed. A new criterion is given for the existence of a diagonal Einstein metric on a nice nilpotent Lie group. Classifications of special classes of Ricci-˛at metrics on nilpotent Lie groups of dimension [eight.tf] are obtained. Some related open questions are presented.

Hardy's Theorem for simply connected nilpotent Lie groups

2001 ◽  
Vol 131 (3) ◽  
pp. 487-494 ◽  
Author(s):  
EBERHARD KANIUTH ◽  
AJAY KUMAR
Keyword(s):  
Fourier Transform ◽  
Lie Groups ◽  
Nilpotent Lie Groups ◽  
Heisenberg Groups ◽  
Hardy’S Theorem ◽  
Simply Connected

We prove an analogue of Hardy's Theorem for Fourier transform pairs in ℝ for arbitrary simply connected nilpotent Lie groups, thus extending earlier work on ℝn and the Heisenberg groups ℍn.

LEFT INVARIANT COMPLEX STRUCTURES ON NILPOTENT SIMPLY CONNECTED INDECOMPOSABLE 6-DIMENSIONAL REAL LIE GROUPS

2007 ◽  
Vol 17 (01) ◽  
pp. 115-139 ◽  
Author(s):  
L. MAGNIN
Keyword(s):  
Automorphism Group ◽  
Normal Form ◽  
Lie Groups ◽  
Lie Algebras ◽  
Lie Group ◽  
Normal Forms ◽  
Equivalence Classes ◽  
Complex Structures ◽  
Simply Connected ◽  
Connected Lie Group

Integrable complex structures on indecomposable 6-dimensional nilpotent real Lie algebras have been computed in a previous paper, along with normal forms for representatives of the various equivalence classes under the action of the automorphism group. Here we go to the connected simply connected Lie group G0 associated to such a Lie algebra 𝔤. For each normal form J of integrable complex structures on 𝔤, we consider the left invariant complex manifold G = (G0, J) associated to G0 and J. We explicitly compute a global holomorphic chart for G and we write down the multiplication in that chart.

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 Equidistribution of dense subgroups on nilpotent Lie groups

2009 ◽  
Vol 30 (1) ◽  
pp. 131-150 ◽  
Author(s):  
EMMANUEL BREUILLARD
Keyword(s):  
Random Walk ◽  
Lie Group ◽  
Analogous Result ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Uniform Measure ◽  
Dense Subgroup ◽  
Symmetric Set ◽  
Dense Subgroups ◽  
Simply Connected

AbstractLet Γ be a dense subgroup of a simply connected nilpotent Lie group G generated by a finite symmetric set S. We consider the n-ball Sn for the word metric induced by S on Γ. We show that Sn (with uniform measure) becomes equidistributed on G with respect to the Haar measure as n tends to infinity. We also prove the analogous result for random walk averages.

Sign in / Sign up

Export Citation Format

Share Document