Dimensional Lie Groups Research Articles

Decomposition of stochastic flows generated by Stratonovich SDEs
 with jumps

Consider a manifold $M$ endowed locally witha pair of complementary distributions $\Delta^H \oplus \Delta^V=TM$ and let$\text{Diff}(\Delta^H,M)$ and $\text{Diff}(\Delta^V, M)$ be the corresponding Lie subgroupsgenerated by vector fields in thecorresponding distributions. We decompose a stochastic flow with jumps, upto astopping time, as $\varphi_t = \xi_t \circ \psi_t$, where $\xi_t \in\text{Diff}(\Delta^H, M)$ and $\psi_t \in \text{Diff}(\Delta^V, M)$. Our mainresult provides Stratonovich stochastic differential equations with jumpsfor each of these two components in the corresponding infinite dimensional Liegroups. We present an extension of the Itô-Ventzel-Kunita formula forstochastic flows with jumps generated by classical Marcus equation (as inKurtz, Pardoux and Protter [11]). The results herecorrespond to anextension ofCatuogno, da Silva and Ruffino[4], where this decomposition was studied for the continuous case.

Projective shape analysis of contours and finite 3D configurations from digital camera images

We analyze infinite dimensional projective shape data collected from digital camera images, focusing on two-sample hypothesis testing for both finite and infinite extrinsic mean configurations. The two sample test methodology is based on a Lie group technique that was derived by Crane and Patrangenaru (J Multivar Anal 102:225–237, 2011) and Qiu et al. (Neighborhood hypothesis testing for mean change on infinite dimensional Lie groups and 3D projective shape analysis of matched contours, 2015). In infinite dimensions, the equality of two extrinsic means is likely to be rejected, thus a neighborhood hypothesis is suitably tested, combining the ideas in these two papers with data analysis methods on Hilbert manifolds in Ellingson et al. (J Multivar Anal 122:317–333, 2013). In this manuscript, we apply these general results to the two sample problem for independent projective shapes of 3D facial configurations and for matched projective shapes of 2D and 3D contours. Digital images of 3D scenes are today at the fingertips of any statistician. Here and in the literature referenced in the in this paper, we provide a methodology for properly analyzing such data, when more pictures of a given scene are available.

3-dimensional left-invariant sub-Lorentzian contact structures

We provide a classification of ts-invariant sub-Lorentzian structures on 3 dimensional contact Lie groups. Our approach is based on invariants arising form the construction of a normal Cartan connection.

Ramification filtrations of certain abelian Lie extensions of local fields

Let G⊂xFq〚x〛 (q is a power of the prime p) be a subset of formal power series over a finite field such that it forms a compact abelian p-adic Lie group of dimension d≥1. We establish a necessary and sufficient condition for the APF extension of local field corresponding to (Fq⸨x⸩,G) under the field of norms functor to be an extension of p-adic fields. We then apply this result to study invertible power series over a ring of p-adic integers which commute with a fixed noninvertible power series under composition.

Normal forms on contracting foliations: smoothness and homogeneous structure

In this paper we consider a diffeomorphism f of a compact manifold \(\mathcal {M}\) which contracts an invariant foliation W with smooth leaves. If the differential of f on TW has narrow band spectrum, there exist coordinates \({\mathcal {H}}_x:W_x\rightarrow T_xW\) in which \(f|_W\) has polynomial form. We present a modified approach that allows us to construct maps \({\mathcal {H}}_x\) that depend smoothly on x along the leaves of W. Moreover, we show that on each leaf they give a coherent atlas with transition maps in a finite dimensional Lie group. Our results apply, in particular, to \(C^1\)-small perturbations of algebraic systems.

Spinor Frenet Equations in Three Dimensional Lie Groups

In this paper, we study spinor Frenet equations in three dimensional Lie Groups with a bi-invariant metric. Also, we obtain spinor Frenet equations for special cases of three dimensional Lie groups.

Gauging without initial symmetry

The gauge principle is at the heart of a good part of fundamental physics: Starting with a group G of so-called rigid symmetries of a functional defined over space-time Sigma, the original functional is extended appropriately by additional Lie(G)-valued 1-form gauge fields so as to lift the symmetry to Maps(Sigma,G). Physically relevant quantities are then to be obtained as the quotient of the solutions to the Euler-Lagrange equations by these gauge symmetries. In this article we show that one can construct a gauge theory for a standard sigma model in arbitrary space-time dimensions where the target metric is not invariant with respect to any rigid symmetry group, but satisfies a much weaker condition: It is sufficient to find a collection of vector fields v_a on the target M satisfying the extended Killing equation v_{a(i;j)}=0 for some connection acting on the index a. For regular foliations this is equivalent to requiring the conormal bundle to the leaves with its induced metric to be invariant under leaf-preserving diffeomorphisms of M, which in turn generalizes Riemannian submersions to which the notion reduces for smooth leaf spaces M/~. The resulting gauge theory has the usual quotient effect with respect to the original ungauged theory: in this way, much more general orbits can be factored out than usually considered. In some cases these are orbits that do not correspond to an initial symmetry, but still can be generated by a finite dimensional Lie group G. Then the presented gauging procedure leads to an ordinary gauge theory with Lie algebra valued 1-form gauge fields, but showing an unconventional transformation law. In general, however, one finds that the notion of an ordinary structural Lie group is too restrictive and should be replaced by the much more general notion of a structural Lie groupoid.

On an invariance property of the space of smooth vectors

Let $(\pi, \mathcal H)$ be a continuous unitary representation of the (infinite dimensional) Lie group $G$ and $\gamma \: \mathbb R \to \mathrm{Aut}(G)$ define a continuous action of $\mathbb R$ on $G$. Suppose that $\pi^\#(g,t) = \pi(g) U_t$ defines a continuous unitary representation of the semidirect product group $G \rtimes_\gamma \mathbb R$. The first main theorem of the present note provides criteria for the invariance of the space $\mathcal H^\infty$ of smooth vectors of $\pi$ under the operators $U_f = \int_\mathbb R f(t)U_t\, dt$ for $f \in L^1(\mathbb R)$, resp., $f \in \mathcal S(\mathbb R)$. Using this theorem we show that, for suitably defined spectral subspaces $\mathfrak g_{\mathbb C}(E)$, $E \subseteq \mathbb R$, in the complexified Lie algebra $\mathfrak g_{\mathbb C}$, and $\mathcal H^\infty(F)$, $F\subseteq \mathbb R$, for $U$ in $\mathcal H^\infty$, we have \[ \mathsf{d}\pi(\mathfrak g_{\mathbb C}(E)) \mathcal H^\infty(F) \subseteq \mathcal H^\infty(E + F).\]

On the left invariant Randers and Matsumoto metrics of Berwald type on 3-dimensional Lie groups

In this paper we identify all simply connected 3-dimensional real Lie groups which admit Randers or Matsumoto metrics of Berwald type with a certain underlying left invariant Riemannian metric. Then we give their flag curvatures formulas explicitly.

On the Links of Simple Singularities, Simple Elliptic Singularities and Cusp Singularities

Abstract This is a survey article about the study of the links of some complex hypersurface singularities in ℂ3 . We study the links of simple singularities, simple elliptic singularities and cusp singularities, and the canonical contact structures on them. It is known that each singularity link is diffeomorphic to a compact quotient of a 3-dimensional Lie group SU (2), Nil3 or Sol3 , respectively. Moreover, the canonical contact structure is equivalent to the contact structure invariant under the action of each Lie group. We show a new proof of this fact using the moment polytope of S5 . Our proof gives a new aspect to the relation between simple elliptic singularities and cusp singularities, and visualizes how the singularity links are embedded in S5 as codimension two contact submanifolds.

Complex, Symplectic, and Contact Structures on Low-Dimensional Lie Groups

It is well known that on any Lie group, a left-invariant Riemannian structure can be defined. For other left-invariant geometric structures, for example, complex, symplectic, or contact structures, there are difficult obstructions for their existence, which have still not been overcome, although a lot of works were devoted to them. In recent years, substantial progress in this direction has been made; in particular, classification theorems for low-dimensional groups have been obtained. This paper is a brief review of left-invariant complex, symplectic, pseudo-Kahlerian, and contact structures on low-dimensional Lie groups and classification results for Lie groups of dimension 4, 5, and 6.

Invariant solutions to the conformal Killing–Yano equation on Lie groups

We search for invariant solutions of the conformal Killing–Yano equation on Lie groups equipped with left invariant Riemannian metrics, focusing on 2-forms. We show that when the Lie group is compact equipped with a bi-invariant metric or 2-step nilpotent, the only invariant solutions occur on the 3-dimensional sphere or on a Heisenberg group. We classify the 3-dimensional Lie groups with left invariant metrics carrying invariant conformal Killing–Yano 2-forms.

Complex structure on the six dimensional sphere from a spontaneous symmetry breaking

Existence of a complex structure on the 6 dimensional sphere is proved in this paper. The proof is based on re-interpreting a hypothetical complex structure as a classical ground state of a Yang–Mills–Higgs-like theory on S6. This classical vacuum solution is then constructed by Fourier expansion (dimensional reduction) from the obvious one of a similar theory on the 14 dimensional exceptional compact Lie group G2.

Generalized harmonic functions and unitary representations of low dimensional Lie groups

The unitary representations of the three dimensional simple Lie groups are reconsidered from the perspective of harmonic functions acting on certain manifolds related to differential realisations of the groups themselves. By means of contractions of Lie groups, the procedure is also applied to the group E2 of rotations-translations in two dimensions.

Associated Curves of Frenet curves in Three Dimensional Compact Lie Group

Associated Curves of Frenet curves in Three Dimensional Compact Lie Group

Invariants of complex structures on nilmanifolds

Let $(N, J)$ be a simply connected $2n$-dimensional nilpotent Lie group endowed with an invariant complex structure. We define a left invariant Riemannian metric on $N$ compatible with $J$ to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. In [7], J. Lauret proved that minimal metrics (if any) are unique up to isometry and scaling. This uniqueness allows us to distinguish two complex structures with Riemannian data, giving rise to a great deal of invariants. We show how to use a Riemannian invariant: the eigenvalues of the Ricci operator, polynomial invariants and discrete invariants to give an alternative proof of the pairwise non-isomorphism between the structures which have appeared in the classification of abelian complex structures on 6-dimensional nilpotent Lie algebras given in [1]. We also present some continuous families in dimension 8.

The diffeomorphism group of a non-compact orbifold

We endow the diffeomorphism group of a paracompact (reduced) orbifold with the structure of an infinite dimensional Lie group modelled on the space of compactly supported sections of the tangent orbibundle. For a second countable orbifold, we prove that this Lie group is C^0-regular and thus regular in the sense of Milnor. Furthermore an explicit characterization of the Lie algebra associated to the diffeomorphism group of an orbifold is given.

Euler-Poincaré equations on semi-direct products

We study the Euler-Poincare equations on the semi-direct products of the group of orientation preserving diffeomorphisms of the circle with itself. We establish a reduction result to the direct product structure which will allow us to investigate the well-posedness, in the smooth category, using geodesic flows on infinite dimensional Lie groups.

HIGHER ORDER CONGRUENCES AMONGST HASSE–WEIL -VALUES

AbstractFor the $(d+1)$-dimensional Lie group $G=\mathbb{Z}_{p}^{\times }\ltimes \mathbb{Z}_{p}^{\oplus d}$, we determine through the use of $p$-power congruences a necessary and sufficient set of conditions whereby a collection of abelian $L$-functions arises from an element in $K_{1}(\mathbb{Z}_{p}\unicode[STIX]{x27E6}G\unicode[STIX]{x27E7})$. If $E$ is a semistable elliptic curve over $\mathbb{Q}$, these abelian $L$-functions already exist; therefore, one can obtain many new families of higher order $p$-adic congruences. The first layer congruences are then verified computationally in a variety of cases.

The classification of totally umbilical surfaces in homogeneous 3-manifolds

We obtain an exhaustive classification of totally umbilical surfaces in unimodular and non-unimodular simply-connected 3-dimensional Lie groups endowed with arbitrary left-invariant Riemannian metrics. This completes the classification of totally umbilical surfaces in homogeneous Riemannian 3-manifolds.