scholarly journals Spherical Harmonics on the Heisenberg Group

1980 ◽  
Vol 23 (4) ◽  
pp. 383-396 ◽  
Author(s):  
Peter C. Greiner
Keyword(s):  
Heisenberg Group ◽  
Spherical Harmonics ◽  
Lie Group ◽  
Nilpotent Lie Group ◽  
Similar Group ◽  
Image Position ◽  
Group Law

H1. Equip ℝ3 with the group law(1.1)where (z, t) stands for (x, y, t). This is a nilpotent Lie group, usually referred to as the first Heisenberg group, H1. In general Hk denotes ℝ2k+1 equipped with a similar group law, namely

scholarly journals Harmonic analysis on the quotient spaces of Heisenberg groups

1991 ◽  
Vol 123 ◽  
pp. 103-117 ◽  
Author(s):  
Jae-Hyun Yang
Keyword(s):  
Quantum Mechanics ◽  
Heisenberg Group ◽  
Harmonic Analysis ◽  
Unitary Representation ◽  
Lie Group ◽  
Theta Series ◽  
Foundations Of Quantum Mechanics ◽  
Nilpotent Lie Group ◽  
Heisenberg Groups ◽  
Quotient Spaces

A certain nilpotent Lie group plays an important role in the study of the foundations of quantum mechanics ([Wey]) and of the theory of theta series (see [C], [I] and [Wei]). This work shows how theta series are applied to decompose the natural unitary representation of a Heisenberg group.

scholarly journals Equidistribution of joinings under off-diagonal polynomial flows of nilpotent Lie groups

2012 ◽  
Vol 33 (6) ◽  
pp. 1667-1708 ◽  
Author(s):  
TIM AUSTIN
Keyword(s):  
Lie Group ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Norm Convergence ◽  
Ergodic Averages ◽  
Multiple Recurrence ◽  
Polynomial Maps ◽  
Image Position ◽  
Polynomial Flows ◽  
Diagonal Subgroup

AbstractLet $G$ be a connected nilpotent Lie group. Given probability-preserving$G$-actions $(X_i,\Sigma _i,\mu _i,u_i)$, $i=0,1,\ldots ,k$, and also polynomial maps $\phi _i:\mathbb {R}\to G$, $i=1,\ldots ,k$, we consider the trajectory of a joining $\lambda $ of the systems $(X_i,\Sigma _i,\mu _i,u_i)$ under the ‘off-diagonal’ flow \[ (t,(x_0,x_1,x_2,\ldots ,x_k))\mapsto (x_0,u_1^{\phi _1(t)}x_1,u_2^{\phi _2(t)}x_2,\ldots ,u_k^{\phi _k(t)}x_k). \] It is proved that any joining $\lambda $ is equidistributed under this flow with respect to some limit joining $\lambda '$. This is deduced from the stronger fact of norm convergence for a system of multiple ergodic averages, related to those arising in Furstenberg’s approach to the study of multiple recurrence. It is also shown that the limit joining $\lambda '$ is invariant under the subgroup of $G^{k+1}$generated by the image of the off-diagonal flow, in addition to the diagonal subgroup.

scholarly journals Generalized Ricci flow on nilpotent Lie groups

2021 ◽  
Vol 33 (4) ◽  
pp. 997-1014
Author(s):  
Fabio Paradiso
Keyword(s):  
Heisenberg Group ◽  
Lie Group ◽  
Ricci Flow ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Geometric Flows ◽  
Courant Algebroid ◽  
New Family ◽  
Lie Brackets ◽  
Simply Connected

Abstract We define solitons for the generalized Ricci flow on an exact Courant algebroid. We then define a family of flows for left-invariant Dorfman brackets on an exact Courant algebroid over a simply connected nilpotent Lie group, generalizing the bracket flows for nilpotent Lie brackets in a way that might make this new family of flows useful for the study of generalized geometric flows such as the generalized Ricci flow. We provide explicit examples of both constructions on the Heisenberg group. We also discuss solutions to the generalized Ricci flow on the Heisenberg group.

scholarly journals On seven-dimensional quaternionic contact solvable Lie groups

2014 ◽  
Vol 26 (2) ◽  
Author(s):  
Diego Conti ◽  
Marisa Fernández ◽  
José A. Santisteban
Keyword(s):  
Heisenberg Group ◽  
Lie Groups ◽  
Lie Group ◽  
Contact Structure ◽  
Contact Manifold ◽  
Nilpotent Lie Group ◽  
Solvable Lie Groups ◽  
Quaternionic Heisenberg Group

AbstractWe answer in the affirmative a question posed by Ivanov and Vassilev on the existence of a seven-dimensional quaternionic contact manifold with closed fundamental 4-form and non-vanishing torsion endomorphism. Moreover, we show an approach to the classification of seven-dimensional solvable Lie groups having an integrable left invariant quaternionic contact structure. In particular, we prove that the unique seven-dimensional nilpotent Lie group admitting such a structure is the quaternionic Heisenberg group.

scholarly journals Transitivity of Heisenberg group extensions of hyperbolic systems

2011 ◽  
Vol 32 (1) ◽  
pp. 223-235 ◽  
Author(s):  
IAN MELBOURNE ◽  
VIOREL NIŢICĂ ◽  
ANDREI TÖRÖK
Keyword(s):  
Dynamical System ◽  
Heisenberg Group ◽  
Lie Group ◽  
Commutator Subgroup ◽  
Hyperbolic Systems ◽  
Nilpotent Lie Group ◽  
Topological Transitivity ◽  
Group Extensions ◽  
Topologically Transitive ◽  
Dimension 2

AbstractWe show that amongCrextensions (r>0) of a uniformly hyperbolic dynamical system with fiber the standard real Heisenberg group ℋnof dimension 2n+1, those that avoid an obvious obstruction to topological transitivity are generically topologically transitive. Moreover, if one considers extensions with fiber a connected nilpotent Lie group with a compact commutator subgroup (for example ℋn/ℤ), among those that avoid the obvious obstruction, topological transitivity is open and dense.

ON THE QUADRATIC HEISENBERG GROUP

2010 ◽  
Vol 13 (04) ◽  
pp. 551-587 ◽  
Author(s):  
LUIGI ACCARDI ◽  
HABIB OUERDIANE ◽  
HABIB REBEÏ
Keyword(s):  
White Noise ◽  
Heisenberg Group ◽  
Lie Group ◽  
Unitary Operator ◽  
Fock Space ◽  
Local Chart ◽  
Rational Expression ◽  
The One ◽  
Interacting Fock Space ◽  
Group Law

In this paper we introduce the quadratic Weyl operators canonically associated to the one mode renormalized square of white noise (RSWN) algebra as unitary operator acting on the one mode interacting Fock space {Γ, {ωn, n ∈ ℕ}, Φ} where {ωn, n ∈ ℕ} is the principal Jacobi sequence of the nonstandard (i.e. neither Gaussian nor Poisson) Meixner classes. We deduce the quadratic Weyl relations and construct the quadratic analogue of the Heisenberg Lie group with one degree of freedom. In particular, we determine the manifold structure of the group and introduce a local chart containing the identity on which the group law has a simple rational expression in the chart coordinates (see Theorem 6.3).

PROPER ACTIONS ON SOME EXPONENTIAL SOLVABLE HOMOGENEOUS SPACES

2005 ◽  
Vol 16 (09) ◽  
pp. 941-955 ◽  
Author(s):  
ALI BAKLOUTI ◽  
FATMA KHLIF
Keyword(s):  
Maximal Subgroup ◽  
Homogeneous Space ◽  
Lie Group ◽  
Homogeneous Spaces ◽  
Intersection Property ◽  
Nilpotent Lie Group ◽  
Proper Actions ◽  
Simply Connected

Let G be a connected, simply connected nilpotent Lie group, H and K be connected subgroups of G. We show in this paper that the action of K on X = G/H is proper if and only if the triple (G,H,K) has the compact intersection property in both cases where G is at most three-step and where G is special, extending then earlier cases. The result is also proved for exponential homogeneous space on which acts a maximal subgroup.

Equidimensional immersions of locally compact groups

1989 ◽  
Vol 105 (2) ◽  
pp. 253-261 ◽  
Author(s):  
K. H. Hofmann ◽  
T. S. Wu ◽  
J. S. Yang
Keyword(s):  
Exponential Function ◽  
Lie Group ◽  
Compact Groups ◽  
Locally Compact ◽  
Lie Group Theory ◽  
Group A ◽  
Image Position ◽  
Analytic Subgroup ◽  
Algebra Level ◽  
Connected Lie Group

Dense immersions occur frequently in Lie group theory. Suppose that exp: g → G denotes the exponential function of a Lie group and a is a Lie subalgebra of g. Then there is a unique Lie group ALie with exponential function exp:a → ALie and an immersion f:ALie→G whose induced morphism L(j) on the Lie algebra level is the inclusion a → g and which has as image an analytic subgroup A of G. The group Ā is a connected Lie group in which A is normal and dense and the corestrictionis a dense immersion. Unless A is closed, in which case f' is an isomorphism of Lie groups, dim a = dim ALie is strictly smaller than dim h = dim H.

On Flag Curvature of Homogeneous Randers Spaces

2013 ◽  
Vol 65 (1) ◽  
pp. 66-81 ◽  
Author(s):  
Shaoqiang Deng ◽  
Zhiguang Hu
Keyword(s):  
Explicit Formula ◽  
Lie Group ◽  
Flag Curvature ◽  
Nilpotent Lie Group ◽  
Finsler Spaces ◽  
Rigidity Result ◽  
Randers Space ◽  
Negatively Curved

AbstractIn this paper we give an explicit formula for the flag curvature of homogeneous Randers spaces of Douglas type and apply this formula to obtain some interesting results. We first deduce an explicit formula for the flag curvature of an arbitrary left invariant Randersmetric on a two-step nilpotent Lie group. Then we obtain a classification of negatively curved homogeneous Randers spaces of Douglas type. This results, in particular, in many examples of homogeneous non-Riemannian Finsler spaces with negative flag curvature. Finally, we prove a rigidity result that a homogeneous Randers space of Berwald type whose flag curvature is everywhere nonzero must be Riemannian.

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