Probability theory on simply connected nilpotent Lie groups

1996 ◽  
pp. 7-27
Author(s):  
Daniel Neuenschwander
Keyword(s):  
Probability Theory ◽  
Lie Groups ◽  
Nilpotent Lie Groups ◽  
Simply Connected

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.

scholarly journals LEFT-INVARIANT ALMOST PARA-COMPLEX EINSTEINIAN STRUCTURES ON SIX-DIMENSIONAL NILPOTENT LIE GROUPS

2017 ◽  
pp. 88-95
Author(s):  
Nikolay Smolentsev ◽  
Nikolay Smolentsev
Keyword(s):  
Lie Groups ◽  
Complex Structure ◽  
New Left ◽  
Riemannian Metrics ◽  
Nilpotent Lie Groups ◽  
Complex Structures ◽  
Structural Group ◽  
Almost Complex ◽  
Flat Structures ◽  
Simply Connected

As is well known, there are 34 classes of isomorphic simply connected six-dimensional nilpotent Lie groups. Of these, only 26 classes admit left-invariant symplectic structures and only 18 admit left-invariant complex structures. There are five six-dimensional nilpotent Lie groups G , which do not admit neither symplectic, nor complex structures and, therefore, can be neither almost pseudo- Kӓhlerian, nor almost Hermitian. In this work, these Lie groups are being studied. The aim of the paper is to define new left-invariant geometric structures on the Lie groups under consideration that compensate, in some sense, the absence of symplectic and complex structures. Weakening the closedness requirement of left-invariant 2-forms ω on the Lie groups, non-degenerated 2-forms ω are obtained, whose exterior differential dω is also non-degenerated in Hitchin sense [6]. Therefore, the Hitchin’s operator K dω is defined for the 3-form dω . It is shown that K dω defines an almost complex or almost para-complex structure for G and the couple ( ω, dω ) defines pseudo-Riemannian metrics of signature (2,4) or (3,3), which is Einsteinian for 4 out of 5 considered Lie groups. It gives new examples of multiparametric families of Einstein metrics of signature (3,3) and almost para-complex structures on six-dimensional nilmanifolds, whose structural group is being reduced to SL (3 , R) SO (3 , 3). On each of the Lie groups under consideration, compatible pairs of left-invariant forms (ω, Ω), where Ω = d ω, are obtained. For them the defining properties of half-flat structures are naturally fulfilled: d Ω = 0 and ωΩ = 0. Therefore, the obtained structures are not only almost Einsteinian para-complex, but also pseudo- Riemannian half-flat.

Changes of Variables Preserving Fourier-Stieltjes Transforms on Simply Connected Nilpotent Lie Groups

1977 ◽  
Vol 29 (4) ◽  
pp. 744-755
Author(s):  
S. W. Drury
Keyword(s):  
Lie Groups ◽  
Nilpotent Lie Groups ◽  
Continuous Map ◽  
Image Position ◽  
Simply Connected ◽  
Stieltjes Transforms

In [1] Beurling and Helson prove the following theorem.THEOREM. Let φ: R → R be a continuous map such thatThen φ is affine. (Here denotes the space of Fourier-Stieltjes transforms on R.)

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 ON THE RANDERS METRICS ON TWO-STEP HOMOGENEOUS NILMANIFOLDS OF DIMENSION FIVE

2011 ◽  
Vol 08 (03) ◽  
pp. 501-510 ◽  
Author(s):  
HAMID REZA SALIMI MOGHADDAM
Keyword(s):  
Scalar Curvature ◽  
Sectional Curvature ◽  
Lie Groups ◽  
Three Dimensional ◽  
Flag Curvature ◽  
Nilpotent Lie Groups ◽  
Randers Metric ◽  
Levi Civita Connection ◽  
Simply Connected ◽  
Randers Metrics

In this paper we study the geometry of simply connected two-step nilpotent Lie groups of dimension five. We give the Levi–Civita connection, curvature tensor, sectional and scalar curvatures of these spaces and show that they have constant negative scalar curvature. Also we show that the only space which admits left-invariant Randers metric of Berwald type has three-dimensional center. In this case the explicit formula for computing flag curvature is obtained and it is shown that flag curvature and sectional curvature have the same sign.

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