Left-invariant Vector Fields Research Articles

Optimal Control on the Heisenberg Group

Let H denote either the Heisenberg group \Bbb R^{2n+1}, or the Cartesian product of n copies of the three-dimensional Heisenberg group \Bbb R^3. Let lX1, Y1, …, Xn, Ynr be an independent set of left-invariant vector fields on H. In this paper, we study the left-invariant optimal control problem on H with the dynamics \dot{q}(t)=\sum\limits^n_{i=1}u_i(t)X_i(q(t))+v_i(t)Y(q(t)), the cost functional \Lambda(q,u)={1 \over 2}\int \sum\limits^n_{i=1}\mu_i(u^2_i+v^2_i), with arbitrary positive parameters μ1, …, μn, and admissible controls taken from the set of measurable functions t\mapsto u(t)=(u_1(t),v_1(t),\ldots , u_n(t), v_n(t)). The above control system is encoded either in the kernel of a contact 1-form (for \Bbb R^{2n+1}), or in the kernel of a Pfaffian system (for \Bbb R^{3n}). In both cases, the action of the semi-direct product of the torus Tn with H describe the symmetries of the problem. The Pontryagin maximum principle provides optimal controlss extremal trajectories are solutions to the Hamiltonian system associated with the problem. Abnormal extremals (which do not depend on the cost functional) yield solutions that are geometrically irrelevant. An explicit integration of the extremal equations provides a tool for studying some aspects of the sub-Riemannian structure defined on H by means of the above optimal control problem.

SU(3) revisited

The `D' matrices for all states of the two fundamental representations and octet are given in the Euler angle parametrization of SU(3). The raising and lowering operators are given in terms of linear combinations of the left-invariant vector fields of the group manifold in this parametrization. Using these differential operators the highest weight state of an arbitrary irreducible representation is found and a description of the calculation of Clebsch-Gordon coefficients is given.

Differential geometry on SU(3) with applications to three state systems

The left and right invariant vector fields are calculated in an “Euler angle”-type parametrization for the group manifold of SU(3), referred to here as Euler coordinates. The corresponding left and right invariant one-forms are then calculated. This enables the calculation of the invariant volume element or Haar measure. These are then used to describe the density matrix of a pure state and geometric phases for three state systems.

A logarithmic Sobolev inequality for the free loop group

For a compact, connected Lie group. G. let L( G) be the corresponding free loop group. For a given Ad inG,-invariant inner product on the Lie algebra of G and a suitable related inner product on a Cameron-Martin subspace of the Lie algebra of L( G) we derive the corresponding Riemannian slructure for left-invariant vector fields on L( G). We then prove a logarithmic Sobolev inequality on L( G)

Estimates for spectral projection operators of the sub-laplacian on the heisenberg group

In this paper, we use Laguerre calculus to find theLP spectrum (λ, Μ) of the pair (L, iT). Here\({\cal L} = - \frac{1}{2}\sum\limits_{j = 1}^n {\left( {Z_j \bar Z_j + \bar Z_j Z_j } \right)} \) md T = ∂/∂t with\(\left\{ {Z_1 ,...,Z_n ,\bar Z_1 ,,...,\bar Z_n , T} \right\}\)a basis for the left-invariant vector fields on the Heisenberg group. We find kernels for the spectral projection operators on the ray λ > 0 in the Heisenberg brush and show that they are Calderon-Zygmund-Mikhlin operators. Estimates for these operators in L k p (Hn), HP(Hn), and S k pv (Hn) spaces can therefore be deduced.

Geometric vector fields on Lie groups

We explore some geometric properties of Lie groups admitting a left-invariant vector field with equidistant flow lines.

On positive entire solutions to the Yamabe-type problem on the Heisenberg and stratified groups

Let G \mathbb {G} be a nilpotent, stratified homogeneous group, and let X 1 X_{1} , …, X m X_{m} be left invariant vector fields generating the Lie algebra G \mathcal {G} associated to G \mathbb {G} . The main goal of this paper is to study the Yamabe type equations associated with the sub-Laplacian Δ G = ∑ k = 1 m X k 2 ( x ) \Delta _{\mathbb {G}} = \sum _{k=1}^m X_k^2(x) on G \mathbb {G} : (*) Δ G u + K ( x ) u p = 0. \begin{equation}\tag {*} \Delta _{\mathbb {G}} u+K(x)u^{p}=0. \end{equation} Especially, we will establish the existence, nonexistence and asymptotic behavior of positive solutions to ( ∗ * ). Our results include the Yamabe type problem on the Heisenberg group as a special case, which is of particular importance and interest and also appears to be new even in this case.

VECTOR FIELDS ON QUANTUM GROUPS

We construct the space of vector fields on a generic quantum group. Its elements are products of elements of the quantum group itself with left-invariant vector fields. We study the duality between vector fields and one-forms and generalize the construction to tensor fields. A Lie derivative along any (also non-left-invariant) vector field is proposed and a puzzling ambiguity in its definition discussed. These results hold for a generic Hopf algebra.

Optimal regularity for quasilinear equations in stratified nilpotent Lie groups and applications

We announce the optimal C1+α regularity of the gradient of weak solutions to a class of quasilinear degenerate elliptic equations in nilpotent stratified Lie groups of step two. As a consequence we also prove a Liouville type theorem for 1-quasiconformal mappings between domains of the Heisenberg group Hn. Statement of the results Consider the quasilinear elliptic equation n ∑ i=1 ∂xiAi(x,∇u) = 0, (1) where Ai(x, ξ) : R → R, i = 1, . . . , n, are differentiable functions satisfying λ|η| ≤ ∑n i,j=1 ∂ξjAi(x, ξ)ηiηj ≤ λ−1|η|2, and ∑n i,j=1 ∂xjAi(x, ξ) ≤ C(1 + |ξ|), for every η ∈ R, and almost every x, ξ ∈ R. The sharp C regularity of weak solutions to (1) is one of the pillars on which the modern theory of quasilinear partial differential equations rests. The ideas on which its proof is based form a recurring theme in nonlinear analysis: first use difference quotients to prove that the weak solutions admit second (weak) derivatives, then differentiate the equations and observe that the derivatives of the solution are themselves solutions to some linear partial differential equations, whose coefficients are not very regular. At this point the regularity theory for linear equations with nonsmooth coefficients provides the final step in the proof of the Holder continuity of the gradient of weak solutions to (1). The observation that one could reduce the study of quasilinear equations to studying linear equations with “bad” coefficients goes back to the pioneering work of Morrey, and has been developed by many mathematicians in the last thirty years. Although (1) is a relatively simple elliptic equation, the regularity theorem has far-reaching applications in calculus of variations (see, for instance, [Gi]) and in the theory of quasiconformal mappings in space [G1]. Two natural and inter-related questions arise: Can the ellipticity hypothesis in (1) be somewhat weakened, and still expect regularity of the gradient of the solutions? Is this “new” problem of some geometric relevance? In this announcement we provide positive answers to both questions. Received by the editors March 15, 1996. 1991 Mathematics Subject Classification. Primary 35H05. Alfred P. Sloan Doctoral Dissertation Fellow. c ©1996 American Mathematical Society 60 OPTIMAL REGULARITY 61 In the proof of his famous rigidity theorem [Mo], Mostow introduced quasiconformal mappings in the setting of stratified nilpotent Lie groups [F]. With this name one refers to the class of simply connected Lie groups G endowed with a stratification of the Lie algebra g = V 1 ⊕ · · · ⊕ V , with r ≥ 1 (the step of the group), such that [V , V j ] = V , j = 1, . . . , r − 1, and [V , V ] = 0. The simplest example of a stratified Lie group is the Euclidean space R, with r = 1. A less trivial, and genuinely non-Euclidean example is provided by the Heisenberg group H, n ≥ 1, whose Lie algebra is h = R ⊕ R. The central role played by the Heisenberg group in many problems of complex geometry, representation theory and partial differential equations makes H the prototype par excellence of stratified nilpotent Lie groups. The theory of quasiconformal mappings between domains of the Heisenberg group has been developed recently in a series of papers by Koranyi and Reimann [KR1]–[KR3] and by Pansu [P]. As in the Euclidean case (see [G1] and [R]), this development led to various questions concerning the regularity of weak solutions to a class of quasilinear equations similar to (1). The notion of quasiconformality is a metric one, and in this setting it is related to the Carnot-Caratheodory metric associated to a basis X 1 , . . . , X 1 m, m = m =dim(V ) of V 1 (with our notation we do not distinguish between elements of g and left invariant vector fields). Since this metric is nonisotropic, it is natural to expect some nonisotropic structure in the relevant equations. In order to be more precise we need to recall that the exponential mapping exp : g → G is a diffeomorphism, and so we can use exponential coordinates p = (p1, . . . , p 1 m, p 2 1, . . . , p 2 dim(V 2), . . . ) on G. Let Xu = (X 1u, . . . ,X 1 mu) denote the horizontal gradient of the function u. Consider the equation

Bicovariant differential geometry of the quantum group SL h (2)

There are only two quantum group structures on the space of two by two unimodular matrices, these are the SL q (2) and the SL h (2) quantum groups. The differential geometry of SL q (2) is well known. In this Letter, we develop the differential geometry of SL h (2), and show that the space of left invariant vector fields is three-dimensional.

The inverse of some differential operators on the heisenberg group

The purpose of this article is to obtain the fundamental solution of differential operators on the Heisenberg group Hn Where denote the standard basis of the Lie algebra hn on left-invariant vector fields on Hn (See the introduction for definition). We find the symbol of the inverse of and the kernel related to this symbol. We also find the formal link between and the Heisenberg Laplacian £α

Compensated compactness and the Heisenberg group

Jacobians of maps on the Heisenberg group are shown to map suitable group Sobolev spaces into the group Hardy space H1. From this result and a weak∗ convergence theorem for the Hardy space H1 of the Heisenberg group, a compensated compactness property for these Jacobians is obtained. 0. Introduction We investigate compensated compactness properties of Jacobians of maps on the Heisenberg group and we prove results analogous to those for the Jacobians of maps on R. Let H = C × R be the Lie group with multiplicative structure (z, t) · (z′, t′) = (z + z′, t+ t′ + 2 Im z · z′) where z = (z1, . . . , zn), z′ = (z′ 1, . . . , z′ n) and z · z′ = ∑n j=1 zj z ′ j . This is the n order Heisenberg group. It can be shown that the group operation is C∞ on the manifold C × R and hence H is a locally compact Lie group. Let (z1, . . . , zn, t) be coordinates on H. Write zj = xj + iyj and define the vector fields: Xj = ∂ ∂xj + 2yj ∂ ∂t Yj = ∂ ∂yj − 2xj ∂ ∂t T = ∂ ∂t . It turns out that the Xj , Yj and T form a basis for the left invariant vector fields on H . They satisfy the commutation relations [Xj , Yj ] = −4T, j = 1, . . . , n and all other ∗Work partially supported by the National Science Foundation. ∗∗Work partially supported by NSF grant DMS-9007491. Mathematics Subject Classification (1991): 42 Typeset by AMS-TEX 1 commutators vanish. The group H is equipped with a natural dilation structure r(z, t) = (rz, rt), r > 0, which is consistent with the group multiplication. The associated norm |(z, t)| = (|z|4 + t) is homogeneous of degree 1 with respect to this group of dilations. We denote by Br(u0) the Heisenberg group ball {u ∈ H : |u−1u0| 0 |(f ∗ φδ)(u)| where φδ(u) = δ−2n−2φ(δ−1u). If γ > 2n+2 2n+3 , the Hardy space H (H) = H is the set of all f such that f ∈ L(H). An alternative definition of H(H) can be given via the atomic decomposition. See [FOS] or [CW] for details. Given a C map F = (f1, . . . , fn) from H into R, define (0.3) Jac(F ) = det   L1f1 L1f2 . . . L1fn L2f1 L2f2 . . . L2fn .. .. .. Lnf1 Lnf2 . . . Lnfn   , where Lj is either Xj or Yj . We would like to show that Jac(F ) maps a product of suitable group Sobolev spaces into the spaces H(H) for 1 ≥ γ > 2n+2 2n+3 . The analogous result for the vector fields ∂ ∂xj , 1 ≤ j ≤ n on R n has been proved by P. L. Lions and Y. Meyer when γ = 1 and extended by [CLMS] for 1 ≥ γ > n n+1 . Note that the lower bound for γ in both cases is d d+1 , where d is the homogeneous dimension of the group. Our first result is Theorem 1. Let n ≥ 2 and for 1 ≤ j ≤ n, let 1 0 that depends only on n and on the pj’s such that for every map F = (f1, . . . fn) on H we have: (0.4) ‖ Jac(F )‖Hγ ≤ C n ∏

On Fock-space representations of quantized enveloping algebras related to noncommutative differential geometry

In this paper explicitly natural (from the geometrical point of view) Fock-space representations (contragradient Verma modules) of the quantized enveloping algebras are constructed. In order to do so, one starts from the Gauss decomposition of the quantum group and introduces the differential operators on the corresponding q-deformed flag manifold (assumed as a left comodule for the quantum group) by a projection to it of the right action of the quantized enveloping algebra on the quantum group. Finally, the representatives of the elements of the quantized enveloping algebra corresponding to the left-invariant vector fields on the quantum group are expressed as first-order differential operators on the q-deformed flag manifold.

On the addition of quantum matrices

An addition law is introduced for the usual quantum matrices A(R) by means of a coaddition Δ_t=t⊗1+1⊗t. It supplements the usual comultiplication Δt=t⊗t and together they obey a codistributivity condition. The coaddition does not form a usual Hopf algebra but a braided one. The same remarks apply for rectangular m×n quantum matrices. As an application, left-invariant vector fields are constructed on A(R) and other quantum spaces. They close in the form of a braided Lie algebra. As another application, the wave functions in the lattice approximation of Kac–Moody algebras and other lattice fields can be added and functionally differentiated.

Maximal estimates for nonsymmetric semigroups

Let $X_0, X_1,..., X_k$ be left-invariant vector fields on a Lie group and let $L = ∑_{i=1}^k X_i^2 + X_0$. Then L is the infinitesimal generator of a semigroup ${p_t}_{t≥0}$ of probability measures on G. Let $P*f(x) = ∑_{0

Integration by parts for a Lie group valued Brownian motion

LetG be a Lie group ofd×d matrices and ζ be theLLie algebra ofG. We choose some Euclidean norm on ζ, and an orthonormal basis (D1,...Dm) relative to it. Let\(\tilde D_1 ,...,\tilde D_m \) be the corresponding left invariant vector fields onG. In this paper we derive an integration by parts formula for aG-valued Brownian motion corresponding to the “Laplacian”\(\sum _1^m \tilde D_i^2 \).

Explicit fundamental solutions for a class of left invariant operators on the nilpotent lie groupG d 1+d 2

In this paper, we give an explicit expression of the fundamental solutions and the global solvability for a class of LPDO's consisting of left invariant vector fields on the nilpotent Lie groupG d 1+d 2.

Index theory and Toeplitz algebras on one-parameter subgroups of Lie groups

We form the Toeplitz C*-algebra 1Γ(G X) associated to the one-parameter subgroup exp(tX) defined by a left-invariant vector field I ona compact Lie group G. We compute the i^-theory of F{G X) and its commutator ideal ^{G X). We also define an abstract analytical index for <T(G X) and show that this analytical index can be computed in terms of topological data. Introduction. Let G be a compact Lie group, let X be a non-zero left-invariant vector field on G, and let \Lχ denote the Lie deriva- tive with respect to X. The operator \Lχ extends to an unbounded self-adjoint operator on let P denote the positive spectral projection of \Lχ. Next, each continuous complex-valued function φ on G gives rise to a bounded operator Mφ on L2(G) via multi- plication. We form a C*-algebra SΓ{G X) that is generated by the set {PMφ : φ e C(G)}, and we call this algebra the Toeplitz algebra of X. In this paper we study SΓ{G X), and in particular, we look at how SΓ{G X) depends on the geometry and topology of G and the choice of X. Our interest in X) stems from two sources. First, several authors have recently obtained very nice results relating Toeplitz al- gebras on flows to the topology of the flows (2, 8). In our case, the vector field X determines a flow on G, and &~{G X) is the Toeplitz algebra of this flow. Another important reason for studying ^{G X) is that this al- gebra contains information about the self-adjoint operator \Lχ. In (4), the authors used Toeplitz algebra techniques to study self-adjoint operators that are elliptic along the leaves of a foliation. In our case, the cosets of exρ(ίX) foliate (?, and \Lχ is elliptic along the leaves of this foliation. In all of the work mentioned above, the authors required their flows and foliations to be minimal. Our foliations are typically not minimal, and in fact can be very far from being minimal. Thus, we must develop techniques for studying ^{G X) that do not require minimality.

Real complexified quantum groups, their vector fields and quantum enveloping algebras

We construct complex quantum groups associated with the Lie algebras of typeA n−1 ,B n ,C n andD n which are considered as real algebras. Following the ideas of Faddeev, Reshetikhin and Takhtayan, we obtain the Hopf algebras of regular functionalsU R , on these real complexified quantum groups. Theq-analogues of the left invariant vector fields of the quantum enveloping algebras are defined. These quantum vector fields are functionals over the corresponding real formA of the complex quantum groupA. The equivalence of the Hopf algebra of regular functionals and the algebra of complex quantum vector fields is shown by factorizing the vector fields uniquely into a triangular and a unitary part and identifying them with the corresponding elements of the algebra of regular functionals. In the special exampleA 1 , we derive theq-deformed real complexified enveloping algebraU q sl(2, ℂ) with six generators.

Vector fields on complex quantum groups

Using previous results we construct theq-analogues of the left invariant vector fields of the quantum enveloping algebra corresponding to the complex Lie algebras of typeA n−1 ,B n ,C n , andD n . These quantum vector fields are functionals over the complex quantum groupA. In the special caseA 1 it is shown that this Hopf algebra coincides withU q sl(2, ℂ).