Lie Subgroup Research Articles

Rolling manifolds on space forms

In this paper, we consider the rolling problem (R) without spinning nor slipping of a smooth connected oriented complete Riemannian manifold (M,g) onto a space form (Mˆ,gˆ) of the same dimension n⩾2. This amounts to study an n-dimensional distribution DR, that we call the rolling distribution, and which is defined in terms of the Levi-Civita connections ∇g and ∇gˆ. We then address the issue of the complete controllability of the control system associated to DR. The key remark is that the state space Q carries the structure of a principal bundle compatible with DR. It implies that the orbits obtained by rolling along loops of (M,g) become Lie subgroups of the structure group of πQ,M. Moreover, these orbits can be realized as holonomy groups of either certain vector bundle connections ∇Rol, called the rolling connections, when the curvature of the space form is non-zero, or of an affine connection (in the sense of Kobayashi and Nomizu, 1996 [14]) in the zero curvature case. As a consequence, we prove that the rolling (R) onto an Euclidean space is completely controllable if and only if the holonomy group of (M,g) is equal to SO(n). Moreover, when (Mˆ,gˆ) has positive (constant) curvature we prove that, if the action of the holonomy group of ∇Rol is not transitive, then (M,g) admits (Mˆ,gˆ) as its universal covering. In addition, we show that, for n even and n⩾16, the rolling problem (R) of (M,g) against the space form (Mˆ,gˆ) of positive curvature c>0, is completely controllable if and only if (M,g) is not of constant curvature c.

On the vertices of indecomposable summands of certain Lefschetz modules

Abstract.We study the reduced Lefschetz module of the complex of

WHEN IS THE DEFORMATION SPACE $\mathscr{T}(\Gamma, H_{2n+1}, H)$ A SMOOTH MANIFOLD?

Let G = H2n + 1 be the 2n + 1-dimensional Heisenberg group and H be a connected Lie subgroup of G. Given any discontinuous subgroup Γ ⊂ G for G/H, a precise union of open sets of the resulting deformation space [Formula: see text] of the natural action of Γ on G/H is derived since the paper of Kobayshi and Nasrin [Deformation of Properly discontinuous action of ℤk and ℝk+1, Internat. J. Math.17 (2006) 1175–1190]. We determine in this paper when exactly this space is endowed with a smooth manifold structure. Such a result is only known when the Clifford–Klein form Γ\G/H is compact and Γ is abelian. When Γ is not abelian or H meets the center of G, the parameter and deformation spaces are shown to be semi-algebraic and equipped with a smooth manifold structure. In the case where Γ is abelian and H does not meet the center of G, then [Formula: see text] splits into finitely many semi-algebraic smooth manifolds and fails to be a Hausdorff space whenever Γ is not maximal, but admits a manifold structure otherwise. In any case, it is shown that [Formula: see text] admits an open smooth manifold as its dense subset. Furthermore, a sufficient and necessary condition for the global stability of all these deformations to hold is established.

On Quadratic Differentials and Twisted Normal Maps of Surfaces in $${\mathbb{S}^{2} \times\mathbb{R}}$$ and $${\mathbb{H}^{2} \times\mathbb{R}}$$

Given a Lie group G with a bi-invariant metric and a compact Lie subgroup K, Bittencourt and Ripoll used the homogeneous structure of quotient spaces to define a Gauss map \({\mathcal{N}:M^{n}\rightarrow{\mathbb{S}}}\) on any hypersupersurface \({M^{n}\looparrowright G/K}\) , where \({{\mathbb{S}}}\) is the unit sphere of the Lie algebra of G. It is proved in Bittencourt and Ripoll (Pacific J Math 224:45–64, 2006) that M n having constant mean curvature (CMC) is equivalent to \({\mathcal{N}}\) being harmonic, a generalization of a Ruh–Vilms theorem for submanifolds in the Euclidean space. In particular, when n = 2, the induced quadratic differential \({\mathcal{Q}_{\mathcal{N}}:=(\mathcal{N}^{\ast}g)^{2,0}}\) is holomorphic on CMC surfaces of G/K. In this paper, we take \({G/K={\mathbb{S}}^{2}\times{\mathbb{R}}}\) and compare \({\mathcal{Q}_{\mathcal{N}}}\) with the Abresch–Rosenberg differential \({\mathcal{Q}}\) , also holomorphic for CMC surfaces. It is proved that \({\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}\) , after showing that \({\mathcal{N}}\) is the twisted normal given by (1.5) herein. Then we define the twisted normal for surfaces in \({{\mathbb{H}}^{2}\times{\mathbb{R}}}\) and prove that \({\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}\) as well. Within the unified model for the two product spaces, we compute the tension field of \({\mathcal{N}}\) and extend to surfaces in \({{\mathbb{H}}^{2}\times{\mathbb{R}}}\) the equivalence between the CMC property and the harmonicity of \({\mathcal{N}.}\)

Isoconstrained Parallel Generators of Schoenflies Motion

Based on the Lie-group-algebraic properties of the displacement set, the 4DOF primitive generators of the Schoenflies motion termed X-motion for brevity are briefly recalled. An X-motion includes 3DOF spatial translation and any 1DOF rotation provided that the axes are parallel to a given direction. The serial concatenation of two generators of 4DOF X-motion produces a 5DOF motion called double Schoenflies motion or X-X-motion for brevity, which includes 3DOFs of translations and any 2DOFs of rotations if the axes are parallel to two independent vectors. This is established using the composition product of two Lie subgroups of X-motion. All possible 5DOF serial chains with distinct general architectures for the generation of X-X-motion are comprehensively introduced in the beginning. The parallel setting between a fixed base and a moving platform of two 5DOF X-X limbs, under particular geometric conditions, makes up a 4DOF isoconstrained parallel generator (abbreviated as IPG-X) of a Schoenflies motion set. “Isoconstrained” is synonymous with “nonoverconstrianed,” and the corresponding chains are trivial chains of the 6D group of general 6DOF motions and can move in the presence of manufacturing errors. Moreover, related families of IPG-Xs are also deducted by using the reordering or the commutation of the factor method, which yields more 5D subsets of displacements containing also the X-motion of the end effector. In that way, several novel general-type architectures of 4DOF parallel manipulators with potential applications are synthesized systematically in consideration of the actuated pairs near the fixed base.

On complex lie supergroups and split homogeneous supermanifolds

It is well known that the category of real Lie supergroups is equivalent to the category of the so-called (real) Harish-Chandra pairs, see [DM], [Kost], [Kosz]. That means that a Lie supergroup depends only on the underlying Lie group and its Lie superalgebra with certain compatibility conditions. More precisely, the structure sheaf of a Lie supergroup and the supergroup morphisms can be explicitly described in terms of the corresponding Lie superalgebra. In this paper we give a proof of this result in the complex-analytic case. Furthermore, if (G, $$ \mathcal{O} $$ G ) is a complex Lie supergroup and H ⊂ G is a closed Lie subgroup, i.e., it is a Lie subsupergroup of (G, $$ \mathcal{O} $$ G ) and its odd dimension is zero, we show that the corresponding homogeneous supermanifold (G/H, $$ \mathcal{O} $$ G/H ) is split. In particular, any complex Lie supergroup is a split supermanifold. It is well known that a complex homogeneous supermanifold may be nonsplit (see, e.g., [OS1]). We find here necessary and sufficient conditions for a complex homogeneous supermanifold to be split.

Filtrations and distortion in infinite-dimensional algebras

A tame filtration of an algebra is defined by the growth of its terms, which has to be majorated by an exponential function. A particular case is the degree filtration used in the definition of the growth of finitely generated algebras. The notion of tame filtration is useful in the study of possible distortion of degrees of elements when one algebra is embedded as a subalgebra in another. A geometric analogue is the distortion of the (Riemannian) metric of a (Lie) subgroup when compared to the metric induced from the ambient (Lie) group. The distortion of a subalgebra in an algebra also reflects the degree of complexity of the membership problem for the elements of this algebra in this subalgebra. One of our goals here is to investigate, mostly in the case of associative or Lie algebras, if a tame filtration of an algebra can be induced from the degree filtration of a larger algebra.

Dimensional upper bounds for admissible subgroups for the metaplectic representation

AbstractWe prove dimensional upper bounds for admissible Lie subgroups H of G = ℍd ⋊ Sp (d, ℝ), d ≥ 2. The notion of admissibility captures natural geometric phenomena of the phase space and it is a sufficient condition for a subgroup to be reproducing. It is expressed in terms of absolutely convergent integrals of Wigner distributions, translated by the affine action of the subgroup. We show that dim H ≤ d2 + 2d, whereas if H ⊂ Sp (d, R), then dim H ≤ d2 + 1. Both bounds are shown to be optimal (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Equivalence relation and partial ordering of parallel mechanisms with applications to mechanism synthesis and mobility analysis

Motion type (or motion pattern) of a mechanism is defined as the set of all rigid motions achievable by the mechanisms’s end-effector; the motion type of a parallel mechanism equals the intersection set of all subchain motion types. The motion type of a non-instantaneous parallel mechanism locally agrees with a regular submanifold (or a Lie subgroup in particular) of the special Euclidean group SE(3). Based on submanifold germs of SE(3), we can define an equivalence relation and a partial order relation for both motion types and parallel mechanisms: two motion types are equivalent if and only if they agree on an open neighborhood around the identity element of SE(3); two motion types are comparable if and only if one is a submanifold of the other on an open neighborhood around the identity of SE(3). It is also possible to define equivalence relation and partial ordering on the collection of parallel mechanisms. In this paper, we first study properties of the equivalence and partial order relation of both motion types and parallel mechanisms, then we discuss their application in type synthesis, mobility analysis and non-overconstrained ness realization of parallel mechanisms.

On groups acting by cohomogeneity one on the euclidean space ℝ n

The paper studies closed Lie subgroups G ⊂ Iso(ℝn) acting by cohomogeneity one on ℝn and prove that when there is no singular orbit, then there is a simply connected, solvable and closed Lie subgroup F ⊂ G which acts by cohomogeneity one on ℝn and the two actions are orbit equivalent.

Symmetry of a symplectic toric manifold

The action of a torus group $T$ on a symplectic toric manifold $(M,\omega)$ often extends to an effective action of a (non-abelian) compact Lie group $G$. We may think of $T$ and $G$ as compact Lie subgroups of the symplectomorphism group $Symp(M,\omega)$ of $(M,\omega)$. On the other hand, $(M,\omega)$ is determined by the associated moment polytope $P$ by the result of Delzant. Therefore, the group $G$ should be estimated in terms of $P$ or we may say that a maximal compact Lie subgroup of $Symp(M,\omega)$ containing the torus $T$ should be described in terms of $P$. In this paper, we introduce a root system $R(P)$ associated to $P$ and prove that any irreducible subsystem of $R(P)$ is of type A and the root system $\Delta(G)$ of the group $G$ is a subsystem of $R(P)$ (so that $R(P)$ gives an upper bound for the identity component of $G$ and any irreducible factor of $\Delta(G)$ is of type A). We also introduce a homomorphism from the normalizer of $T$ in $G$ to an automorphism group $Aut(P)$ of $P$, which detects the connected components of $G$. Finally we find a maximal compact Lie subgroup $G_{\max}$ of $Symp(M,\omega)$ containing the torus $T$.

Deforming discontinuous subgroups for threadlike homogeneous spaces

Let G be an exponential solvable Lie group and H a connected Lie subgroup of G. Given any discontinuous subgroup Γ for the homogeneous space \({\fancyscript{M}=G/H}\) and any deformation of Γ, the deformed discrete subgroup may utterly destroy its proper discontinuous action on \({\fancyscript{M}}\) as H is not compact (except the case when it is trivial). To understand this specific issue, we provide an explicit description of the parameter and the deformation spaces of any abelian discrete Γ acting properly discontinuously and fixed point freely on G/H for an arbitrary H of a threadlike nilpotent Lie group G. The topological features of deformations, such as the local rigidity and the stability are also discussed. Whenever the Clifford-Klein form Γ\G/H in question is assumed to be compact, these spaces are cutely determined and unlike the case of Heisenberg groups, the deformation space fails in general to be a Hausdorff space. We show further that this space admits a smooth manifold as its open dense subset.

The tunneling solutions of the time-dependent Schrödinger equation for a square-potential barrier

The exact tunneling solutions of the time-dependent Schrödinger equation with a square-potential barrier are derived using the continuous symmetry group GS for the partial differential equation. The infinitesimal generators and the elements for GS are represented and derived in the jet space. There exist six classes of wave functions. The representative (canonical) wave functions for the classes are labeled by the eigenvalue sets, whose elements arise partially from the reducibility of a Lie subgroup GLS of GS and partially from the separation of variables. Each eigenvalue set provides two or more time scales for the wave function. The ratio of two time scales can act as the duration of an intrinsic clock for the particle motion. The exact solutions of the time-dependent Schrödinger equation presented here can produce tunneling currents that are orders of magnitude larger than those produced by the energy eigenfunctions. The exact solutions show that tunneling current can be quantized under appropriate boundary conditions and tunneling probability can be affected by a transverse acceleration.

On bundles of covelocities

For a Weil algebra A = \( \mathbb{D}\) r k /I = ℝ ⊕ N A and a manifold M satisfying dimM = m ≥ k, the coincidence of the space T A *M of A-covelocities T x A f: T x A M → T 0 A ℝ with the bundle of the r-th order covelocities T r *M is proved. For a Lie subgroup G A ⊆ G m r of I-preserving \( \mathbb{D}\) m r -automorphisms and a Lie group homomorphism p: G m r → G A it is proved that the space T V,p A *M of T x A f restricted to individual regular p(G m r )-orbits on T m r → M together with the extensions to other regular p(G m r )-orbits coincides with the natural bundle P r M[N A , l] with the standard fiber N A and the left action l: G m r × N A → N A induced by p.

An Embedding Theorem for Automorphism Groups of Cartan Geometries

We study the automorphism group of a Cartan geometry, and prove an embedding theorem analogous to a result of Zimmer for automorphism groups of G-structures. Our embedding theorem leads to general upper bounds on the real rank or nilpotence degree of a Lie subgroup of the automorphism group. We prove that if the maximal real rank is attained in the automorphism group of a geometry of parabolic type, then the geometry is flat and complete.

Type synthesis of primitive Schoenflies-motion generators

A noteworthy type of motion called Schoenflies motion and often termed X-motion for brevity is presented. A specified set of X-motions is endowed with the algebraic structure of a four-dimensional (4D) Lie group. This 4D displacement Lie subgroup includes any translation and any rotation provided that the axis of rotation is parallel to a given direction. In the paper, some preliminary fundamentals about the Lie group of displacements are recalled; the 4D Lie subgroup of X-motion is emphasized. Then serial concatenations of one-dof Reuleaux pairs and hinged parallelograms lead to the enumeration of all possible general architectures of mechanical generators for a given X subgroup. Meanwhile, their corresponding embodiments are graphically displayed for a future use in the structural synthesis of parallel manipulators. These generators are sorted into four classes based on the number of prismatic pairs. In total, forty-three distinct mechanical generators of X-motion are revealed and eighty-two ones having at least one parallelogram are also derived from them. Some chains that are defective generators of X-motion are also identified through an approach based on the group dependency.

Localization for a Matrix-valued Anderson Model

We study localization properties for a class of one-dimensional, matrix-valued, continuous, random Schr\odinger operators, acting on $L^2(\R)\otimes \C^N$, for arbitrary $N\geq 1$. We prove that, under suitable assumptions on the F\urstenberg group of these operators, valid on an interval $I\subset \R$, they exhibit localization properties on $I$, both in the spectral and dynamical sense. After looking at the regularity properties of the Lyapunov exponents and of the integrated density of states, we prove a Wegner estimate and apply a multiscale analysis scheme to prove localization for these operators. We also study an example in this class of operators, for which we can prove the required assumptions on the F\urstenberg group. This group being the one generated by the transfer matrices, we can use, to prove these assumptions, an algebraic result on generating dense Lie subgroups in semisimple real connected Lie groups, due to Breuillard and Gelander. The algebraic methods used here allow us to handle with singular distributions of the random parameters.

A geometric proof of the Karpelevich-Mostow theorem

In this paper we give a geometric proof of the Karpelevich's theorem that asserts that a semisimple Lie subgroup of isometries, of a symmetric space of non compact type, has a totally geodesic orbit. In fact, this is equivalent to a well-known result of Mostow about existence of compatible Cartan decompositions.

Structural Shakiness of Nonoverconstrained Translational Parallel Mechanisms With Identical Limbs

One important category of parallel mechanisms is the translational parallel mechanism (TPM). This paper focuses on the structural shakiness of the non overconstrained TPM. Such a structural shakiness is due to the unavoidable lack of rigidity of the real bodies, which leads to uncheckable orientation changes of the moving platform of a TPM. Using algebraic properties of displacement subsets and, especially, displacement Lie subgroup theory, we show that the structural shakiness of the non overconstrained TPM is inherently determined by the structural type of its limb chains. A structural shakiness index (SSI) for a non overconstrained TPM is introduced. When the set of feasible displacements of the end body of a 5-degree-of-freedom (DOFs) limb chain contains two infinities of parallel axes of rotation, we have SSI = 2; when the displacement set of the end body of a 5-DOF limb chain contains only one infinity of parallel axes of rotation, we have SSI = 1. It is proven that non over con stained TPMs constructed with limb chains with SSI = 1 are much less prone to orientation changes than those constructed with limb chains with SSI = 2. Based on the SSI, we enumerate limb kinematic chains and construct 21 non overconstrained TPMs with less shakiness.

Uncoupled actuation of overconstrained 3T-1R hybrid parallel manipulators

SUMMARYBased on the Lie-group-algebraic properties of the displacement set and intrinsic coordinate-free geometry, several novel 4-dof overconstrained hybrid parallel manipulators (HPMs) with uncoupled actuation of three spatial translations and one rotation (3T-1R) are proposed. In these HPMs, three limbs are those of Cartesian translational parallel mechanisms (CTPMs) and the fourth limb includes an Oldham-type constant velocity shaft coupling (CVSC). The Lie subgroup of Schoenflies (X) displacements of the displacement Lie group and its mechanical generators with nine categories of their general architectures are recalled. A comprehensive enumeration of all possible Oldham-type CVSC limbs is derived fromX-motion generators. Their constant velocity (CV) transmissions are verified by group-algebraic approach. Then, combining one CTPM and one CVSC, we synthesize a lot of uncoupled 3T-1R overconstrained HPMs, which are classified into nine distinct classes of general architectures. In addition, all possible architectures with at least one hinged parallelogram or with one cylindrical pair are disclosed too. At last, related non-overconstrained HPMs are attained by the addition of one idle pair in each limb of the previous HPMs.