幂零Lie群上的不定性原理

设N是具有平方可积表示的幂零Lie群,是其Plancherel测度．本文将N上群Fourier变换矩阵化,并由此给出N上不定性原理的一种定量描述．此外,还对N上不定性原理的定性描述（简称QUP）作了讨论,结果显示出N上QUP与P（λ）的零点集之代数、几何性质的一些联系.     

Hypoelliptic Heat Kernels on Nilpotent Lie Groups

We study property RD in terms of rapid decay of matrix coefficients. We give new formulations of property RD in terms of a L¹-integrability condition of a Banach representation. Combining this new definition with the existence of cyclic subgroups of exponential growth in non-uniform lattices in semisimple Lie groups, we deduce that there exist matrix coefficients associated to several kinds of quasi-regular representations which satisfy a “non-RD condition” for non-uniform lattices. We obtain also that such coefficients can not satisfy the weak inequality of Harish-Chandra.     

Lorentz Geometry of 2-Step Nilpotent Lie Groups

We study the geometry of 2-step nilpotent Lie groups endowed with left-invariant Lorentz metrics. After integrating explicitly the geodesic equations, we discuss the problem of the existence of translated geodesics in those groups. A good part of the paper focuses on the existence of closed timelike geodesics in compact Lorentz 2-step nilmanifolds. Other related results, corollaries, and examples are also presented.     

Two-Step Nilpotent Lie Groups and Homogeneous Fiber Bundles

In this paper, we construct two-step nilpotent Lie groups from homogeneous fiber bundles over compact symmetric spaces. The structure of the constructed nilpotent groups is expressed in terms of the compact Lie groups involved in the fiber bundles. There are close relations between the geometric properties of the nilpotent groups and the total spaces of the fiber bundles. We will find new examples of nilpotent groups which are weakly symmetric and Riemannian geodesic orbit spaces.     

Powers of Sub-Laplacian on Step Two Nilpotent Lie Groups

We give an explicit, and geometrical formula for the fundamental solution for higher order sub-Laplacians on a model step two nilpotent Lie group.     

Nilpotient Lie群上Herz空间及它的某些应用

§1 .　IntroductionLetGbeaconnectedandsimplyconnectednilpotentLiegroupwith(bi invariant)HaarmeasuredgandLiealgebraG .Theexponetialmapissurjectiveby [12 ],Theorem 3.6 .1.Onecanassociatedasubellipticdistance (g ,h)d′( g ;h)witheachfixedalgeraicbasisa1 ,a2 ,…ad′ofG .Foralli∈ { 1,2 ,… ,d′} ,letAi =dL(ai)denotethegeneratorsoflefttranslationsactingontheclass{ai} .Thisdistancehasthecharacterizationd′( g ;h) =sup{ ｜ψ( g) - ψ(h) ｜ :ψ∈C∞0 (G) ,∑d′i=1｜ (Aiψ) ｜2 ≤ 1} ,(see [9],…     

Hypoelliptic Heat Kernel Over 3-Step Nilpotent Lie Groups

In this paper, we provide explicitly the connection between the hypoelliptic heat kernel for some 3-step sub-Riemannian manifolds and the quartic oscillator. We study the left-invariant sub-Riemannian structure on two nilpotent Lie groups, namely, the (2,3,4) group (called the Engel group) and the (2,3,5) group (called the Cartan group or the generalized Dido problem). Our main technique is noncommutative Fourier analysis, which permits us to transform the hypoelliptic heat equation into a one-dimensional heat equation with a quartic potential.     

Inequalities and Local Uncertainty Principles for Nilpotent Lie Groups

Mathematical Notes - Let $$\mathbb{F}_{q}$$ be a finite field, let $$\mathbb{X}$$ be a subset of the projective space ?s?1 over $$\mathbb{F}_{q}$$ parametrized by rational functions,...     

Relatively Free Nilpotent Torsion-Free Groups and Their Lie Algebras

Let K be a field of characteristic zero. For a torsion-free finitely generated nilpotent group G, we naturally associate four finite dimensional nilpotent Lie algebras over K, ? _K (G), grad^(?)(? _K (G)), grad^(g)(exp ? _K (G)), and L _K (G). Let 𝔗 _c be a torsion-free variety of nilpotent groups of class at most c. For a positive integer n, with n ≥ 2, let F _n (𝔗 _c ) be the relatively free group of rank n in 𝔗 _c . We prove that ? _K (F _n (𝔗 _c )) is relatively free in some variety of nilpotent Lie algebras, and ? _K (F _n (𝔗 _c )) ? L _K (F _n (𝔗 _c )) ? grad^(?)(? _K (F _n (𝔗 _c ))) ? grad^(g)(exp ? _K (F _n (𝔗 _c ))) as Lie algebras in a natural way. Furthermore, F _n (𝔗 _c ) is a Magnus nilpotent group. Let G ₁ and G ₂ be torsion-free finitely generated nilpotent groups which are quasi-isometric. We prove that if G ₁ and G ₂ are relatively free of finite rank, then they are isomorphic. Let L be a relatively free nilpotent Lie algebra over ? of finite rank freely generated by a set X. Give on L the structure of a group R, say, by means of the Baker–Campbell–Hausdorff formula, and let H be the subgroup of R generated by the set X. We show that H is relatively free in some variety of nilpotent groups; freely generated by the set X, H is Magnus and L ? ?_?(H) ? L _?(H) as Lie algebras. For relatively free residually torsion-free nilpotent groups, we prove that ? _K and L _K are isomorphic as Lie algebras. We also give an example of a finitely generated Magnus nilpotent group G, not relatively free, such that ?_?(G) is not isomorphic to L _?(G) as Lie algebras.     

s-Stable Laws in Insurance and Finance and Generalization to Nilpotent Lie Groups

s-stable laws on Hilbert spaces, associated with some nonlinear transformations, were introduced by Jurek.^{(16, 18)} Here, we interpret certain s-stable motions as limits of total amount of claims processes (up to a deterministic reserve) of a portfolio of (nontraded) excess-of-loss reinsurance contracts and show that they lead to Erlang's model. We also give explicit formulas for the price of perpetual American options in case the logarithm of the price of the underlying asset is an s-stable motion. Furthermore, we generalize the concept of s-stability to simply connected nilpotent Lie groups. For step 2-nilpotent Lie groups we characterize the Lévy measure and the s-domain of attraction of nongaussian s-stable convolution semigroups.     

具有交换幂零根基的完备李代数的自同构群

给出了复数域上具有交换幂零根基的完备李代数的自同构群．     

Conjugate Loci of Pseudo-Riemannian 2-Step Nilpotent Lie Groups with Nondegenerate Center

We determine the complete conjugate locus along all geodesics parallel or perpendicular to the center (Theorem 2.3). When the center is one-dimensional we obtain formulas in all cases (Theorem 2.5), and when a certain operator is also diagonalizable these formulas become completely explicit (Corollary 2.7). These yield some new information about the smoothness of the pseudoriemannian conjugate locus. We also obtain the multiplicities of all conjugate points.Mathematics Subject Classifications (2000): primary 53C50; secondary 22E25, 53B30, 53C30.     

Spectral Properties of Second Order Differential Operators on Two-step Nilpotent Lie Groups

In this paper, spectral properties of certain left invariant differential operators on two-step nilpotent Lie groups are completely described by using the theory of unitary irreducible representations and the Plancherel formulae on nilpotent Lie groups.     

The Fourier Transform and Its Weyl Symbol on Two-step Nilpotent Lie Groups

In this paper, we give all equivalence classes of irreducible unitary representations for the group H_n ⊗ R^m thereby formulate the Fourier transform on H_n ⊗ R^m (n ≥ 0, m ≥ 0}, which naturally unifies the Fourier transform between the Euclidean group and the Heisenberg group, more generally, between Abelian groups and two-step nilpotent Lie groups. Moreover, by the Plancberel formula for H_n ⊗ R^m we produce the Weyl symbol association with functions of the harmonic oscillator so that to derive the heat kernel on H_n ⊗ R^m.     

Coadjoint Orbits of Three-Step Free Nilpotent Lie Groups and Time-Optimal Control Problem

Doklady Mathematics - We describe coadjoint orbits for three-step free nilpotent Lie groups. It turns out that two-dimensional orbits have the same structure as coadjoint orbits of the Heisenberg...     

$$L^p$$ L p -Multipliers Sensitive to the Group Structure on Nilpotent Lie Groups

Journal of Fourier Analysis and Applications - We propose new sufficient conditions for $$L^p$$ -multipliers on homogeneous nilpotent groups. The multipliers generalise the flag multipliers of...     

Characterization of Shift-Invariant Spaces on a Class of Nilpotent Lie Groups with Applications

Given a simply connected nilpotent Lie group having unitary irreducible representations that are square-integrable modulo the center, we use operator-valued periodization to give a range-function type characterization of shift-invariant spaces of function on the group. We then give characterizations of frame and Riesz families for shift-invariant spaces.     

Nilpotent Lie and Leibniz Algebras

We extend results on finite dimensional nilpotent Lie algebras to Leibniz algebras and counterexamples to others are found. One generator algebras are used in these examples and are investigated further.