Underlying Lie Group Research Articles

Beurling-Fourier algebras on Lie groups and their spectra

We investigate Beurling-Fourier algebras, a weighted version of Fourier algebras, on various Lie groups focusing on their spectral analysis. We will introduce a refined general definition of weights on the dual of locally compact groups and their associated Beurling-Fourier algebras. Constructions of nontrivial weights will be presented focusing on the cases of representative examples of Lie groups, namely SU(n), the Heisenberg group H, the reduced Heisenberg group Hr, the Euclidean motion group E(2) and its simply connected cover E˜(2). We will determine the spectrum of Beurling-Fourier algebras on each of the aforementioned groups emphasizing its connection to the complexification of underlying Lie groups. We also demonstrate that “polynomially growing” weights do not change the spectrum and show the associated regularity of the resulting Beurling-Fourier algebras.

Analytic Torsion, Dynamical Zeta Function, and the Fried Conjecture for Admissible Twists

We show an equality between the analytic torsion and the absolute value at zero of the Ruelle dynamical zeta function on a closed odd dimensional locally symmetric space twisted by an acyclic flat vector bundle obtained by the restriction of a representation of the underlying Lie group. This generalises author’s previous result for unitarily flat vector bundles, and the results of Bröcker, Müller, and Wotzke on closed hyperbolic manifolds.

Geometric Algorithms for Robot Dynamics: A Tutorial Review

Abstract We provide a tutorial and review of the state-of-the-art in robot dynamics algorithms that rely on methods from differential geometry, particularly the theory of Lie groups. After reviewing the underlying Lie group structure of the rigid-body motions and the geometric formulation of the equations of motion for a single rigid body, we show how classical screw-theoretic concepts can be expressed in a reference frame-invariant way using Lie-theoretic concepts and derive recursive algorithms for the forward and inverse dynamics and their differentiation. These algorithms are extended to robots subject to closed-loop and other constraints, joints driven by variable stiffness actuators, and also to the modeling of contact between rigid bodies. We conclude with a demonstration of how the geometric formulations and algorithms can be effectively used for robot motion optimization.

Fréchet Globalisations of Harish-Chandra Supermodules

For any Lie supergroup whose underlying Lie group is reductive, we prove an extension of the Casselman-Wallach globalisation theorem: There is an equivalence between the category of Harish-Chandra modules and the category of SF-representations (smooth Fr\'echet representations of moderate growth) whose module of finite vectors is Harish-Chandra. As an application, we extend to Lie supergroups a general general form of the Gel'fand-Kazhdan criterion due to Sun-Zhu.

On extensions of sub-Riemannian structures on Lie groups

We define the extension of a left-invariant sub-Riemannian structure in terms of an extension of the underlying Lie group and compatibility of the respective distributions and metrics. We show that geodesics of a structure can be lifted to geodesics of any extension of the structure. In the case of central extensions, we show that the normal geodesics of the minimal extension are the projection (in a sense) of the normal geodesics of any other compatible extension. Several illustrative examples are discussed.

On homogeneous Einstein (α,β)-metrics

In this paper, we study homogeneous Einstein ( α , β ) -metrics. First, we deduce a formula for Ricci curvature of a homogeneous ( α , β ) -metric. Based on this formula, we obtain a sufficient and necessary condition for a compact homogeneous ( α , β ) -metric to be Einstein and with vanishing S-curvature. Moreover, we prove that any homogeneous Ricci flat ( α , β ) space with vanishing S-curvature must be a Minkowski space. Finally, we consider left invariant Einstein ( α , β ) -metrics on Lie groups with negative Ricci constant. Under some appropriate conditions, we show that the underlying Lie groups must be two step solvable. We also present a more convenient sufficient and necessary condition for the metric to be Einstein in this special case.

A Finite Dimensional Integrable System Arising in the Study of Shock Clustering

In this work, we consider a finite dimensional Hamiltonian system that contains as a special case an exact discretization of the Lax equation for shock clustering. We characterize the generic coadjoint orbits of the underlying Lie group and establish the Liouville integrability of the system on such orbits. We also solve the Hamiltonian equation explicitly via Riemann–Hilbert factorization problems.

Infinitesimal supersymmetries over Lie groups and their classification for gl(1,1) and sl(1,1)

Infinitesimal supersymmetries over classical Lie groups that do not necessarily integrate to Lie supergroups are described. They yield a notion of supersymmetry that is less rigid than the assumption of a Lie supergroup action but still implies an underlying action of a Lie group. In contrast to Lie supergroups, the arising representation-theoretical Lie supergroups (RTLSG) occur as families associated to Harish-Chandra superpairs. However morphisms of RTLSGs directly correspond to morphisms of Harish-Chandra superpairs. Particular RTLSGs can be derived from the explicit constructions of Lie supergroups given by Berezin and Kostant. The Lie superalgebras $\mathfrak{gl}(1,1)$ or $\mathfrak{sl}(1,1)$ appearing also in higher dimensional classical Lie superalgebras, provide interesting first examples of RTLSGs. A classification of RTLSGs associated to real and complex $\mathfrak{gl}(1,1)$- and $\mathfrak{sl}(1,1)$-Harish-Chandra superpairs is given by parameter spaces and complete sets of invariants. The underlying Lie group is assumed to be connected but possibly not simply connected.

Observers for invariant systems on Lie groups with biased input measurements and homogeneous outputs

This paper provides a new observer design methodology for invariant systems whose state evolves on a Lie group with outputs in a collection of related homogeneous spaces and where the measurement of system input is corrupted by an unknown constant bias. The key contribution of the paper is to study the combined state and input bias estimation problem in the general setting of Lie groups, a question for which only case studies of specific Lie groups are currently available. We show that any candidate observer (with the same state space dimension as the observed system) results in non-autonomous error dynamics, except in the trivial case where the Lie-group is Abelian. This precludes the application of the standard non-linear observer design methodologies available in the literature and leads us to propose a new design methodology based on employing invariant cost functions and general gain mappings. We provide a rigorous and general stability analysis for the case where the underlying Lie group allows a faithful matrix representation. We demonstrate our theory in the example of rigid body pose estimation and show that the proposed approach unifies two competing pose observers published in prior literature.

Ricci curvatures of left invariant Finsler metrics on Lie groups

Left invariant Finsler metrics on Lie groups provide an important class of Finsler manifolds. In this paper, we prove several properties of the Ricci curvatures of such spaces. For example, if all the Ricci curvatures are nonnegative, then the underlying Lie group must be unimodular. If the Lie group is noncommutative and nilpotent, then there must be three directions whose Ricci curvature is positive, negative and zero, respectively. This result gives a negative answer to a question of S.-S. Chern.

THE SHMUSHKEVICH METHOD FOR HIGHER SYMMETRY GROUPS OF INTERACTING PARTICLES

About 60 years ago, I. Shmushkevich presented a simple ingenious method for computing the relative probabilities of channels involving the same interacting multiplets of particles, without the need to compute the Clebsch-Gordan coefficients. The basic idea of Shmushkevich is “isotopic non-polarization” of the states before the interaction and after it. Hence his underlying Lie group was <em>SU</em>(2). We extend this idea to any simple Lie group. This paper determines the relative probabilities of various channels of scattering and decay processes following from the invariance of the interactions with respect to a compact simple a Lie group. Aiming at the probabilities rather than at the Clebsch-Gordan coefficients makes the task easier, and simultaneous consideration of all possible channels for given multiplets involved in the process, makes the task possible. The probability of states with multiplicities greater than 1 is averaged over. Examples with symmetry groups <em>O</em>(5), <em>F</em>(4), and <em>E</em>(8) are shown.

Hamiltonian structures of dynamics of a gyrostat in a gravitational field

The dynamics of a gyrostat in a gravitational field is a fundamental problem in celestial mechanics and space engineering. This paper investigates this problem in the framework of geometric mechanics. Based on the natural symplectic structure, non-canonical Hamiltonian structures of this problem are derived in different sets of coordinates of the phase space. These different coordinates are suitable for different applications. Corresponding Poisson tensors and Casimir functions, which govern the phase flow and phase space structures of the system, are obtained in a differential geometric method. Equations of motion, as well as expressions of the force and torque, are derived in terms of potential derivatives. We uncover the underlying Lie group framework of the problem, and we also provide a systemic approach for equations of motion. By assuming that the gravitational field is axis-symmetrical and central, SO(2) and SO(3) symmetries are introduced into the general problem respectively. Using these symmetries, we carry out two reduction processes and work out the Poisson tensors of the reduced systems. Our results in the central gravitational filed are in consistent with previous results. By these reductions, we show how the symmetry of the problem affects the phase space structures. The tools of geometric mechanics used here provide an access to several powerful techniques, such as the determination of relative equilibria on the reduced system, the energy-Casimir method for determining the stability of equilibria, the variational integrators for greater accuracy in the numerical simulation and the geometric control theory for control problems.

Invariant integration on orthosymplectic and unitary supergroups

The orthosymplectic supergroup OSp(m|2n) and unitary supergroup U(p|q) are studied following a new approach that starts from Harish-Chandra pairs and links the sheaf-theoretical supermanifold approach of Berezin and others with the differential geometry approach of Rogers and others. The matrix elements of the fundamental representation of the Lie supergroup are expressed in terms of functions on the product supermanifold , with the underlying Lie group and N the odd dimension of . This product supermanifold is isomorphic to the supermanifold of . This leads to a new expression for the standard generators of the corresponding Lie superalgebra as invariant derivations on . Using these results, a new and transparent formula for the invariant integrals on OSp(m|2n) and U(p|q) is obtained.

Moduli of Einstein and non-Einstein nilradicals

A nilpotent Lie algebra is called an Einstein nilradical if the corresponding Lie group admits a left-invariant Ricci soliton metric. While these metrics are of independent interest, their existence is intimately related to the existence of Einstein metrics on solvable Lie groups. In this note we are concerned with the following question: How are the Einstein and non-Einstein nilradicals distributed among nilpotent Lie algebras? A full answer to this question is not known and we restrict to the class of 2-step nilpotent Lie groups. Within this class, it is known that a generic group admits a Ricci soliton metric. Using techniques from Geometric Invariant Theory, we study the set of non-generic algebras to learn more about the distribution of non-Einstein nilradicals. Many new (continuous) families of non-isomorphic, non-Einstein nilradicals are constructed. Moreover, the dimension of these families can be arbitrarily large (depending on the dimension of the underlying Lie group). To show such large classes of Lie groups are pairwise non-isomorphic, a new technique is developed to distinguish between Lie algebras.

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.

On discretization of tori of compact simple Lie groups

Three types of numerical data are provided for simple Lie groups of any type and rank. These data are indispensable for Fourier-like expansions of multidimensional digital data into finite series of C- or S-functions on the fundamental domain F of the underlying Lie group G. Firstly, we determine the number |FM| of points in F from the lattice P∨M, which is the refinement of the dual weight lattice P∨ of G by a positive integer M. Secondly, we find the lowest set ΛM of dominant weights, specifying the maximal set of C- and S-functions that are pairwise orthogonal on the point set FM. Finally, we describe an efficient algorithm for finding, on the maximal torus of G, the number of conjugate points to every point of FM. Discrete C- and S-transforms, together with their continuous interpolations, are presented in full generality.

Covariant quantization: spectral analysis versus deformation theory

Formal deformation or rather symbolic calculus? To which extent do these approaches complete each other in the study of symmetry-preserving quantization procedures for homogeneous spaces? The representation theory of underlying Lie groups shows that the answer is much more delicate than initially thought and that it cannot be always reduced to asymptotic expansions with respect to some Planck’s constant. The main goal of this survey is to give hints regarding the aims of each approach and, on the domain where these intersect, to compare the answers they lead to.

$L^2$-homology for compact quantum groups

A notion of $L^2$-homology for compact quantum groups is introduced, generalizing the classical notion for countable, discrete groups. If the compact quantum group in question has tracial Haar state, it is possible to define its $L^2$-Betti numbers and Novikov-Shubin invariants/capacities. It is proved that these $L^2$-Betti numbers vanish for the Gelfand dual of a compact Lie group and that the zeroth Novikov-Shubin invariant equals the dimension of the underlying Lie group. Finally, we relate our approach to the approach of A. Connes and D. Shlyakhtenko by proving that the $L^2$-Betti numbers of a compact quantum group, with tracial Haar state, are equal to the Connes-Shlyakhtenko $L^2$-Betti numbers of its Hopf $*$-algebra of matrix coefficients.

Algorithms for Computations in Local Symmetric Spaces

In the last two decades much of the algebraic/combinatorial structure of Lie groups, Lie algebras, and their representations has been implemented in several excellent computer algebra packages, including LiE, GAP4, Chevie, Magma, and Maple. The structure of reductive symmetric spaces or more generally symmetric k-varieties is very similar to that of the underlying Lie group, with a few additional complications. A computer algebra package enabling one to do computations related to these symmetric spaces would be an important tool for researchers in many areas of mathematics, including representation theory, Harish Chandra modules, singularity theory, differential and algebraic geometry, mathematical physics, character sheaves, Lie theory, etc. In this article we lay the groundwork for computing the fine structure of symmetric spaces over the real numbers and other base fields, give a complete set of algorithms for computing the fine structure of symmetric varieties and use this to compute nice bases for the local symmetric varieties.

Computing the fine structure of real reductive symmetric spaces

Much of the structure of Lie groups has been implemented in several computer algebra packages, including LiE, GAP4, Chevie, Magma and Maple. The structure of reductive symmetric spaces is very similar to that of the underlying Lie group and a computer algebra package for computations related to symmetric spaces would be an important tool for researchers in many areas of mathematics. Until recently only very few algorithms existed for computations in symmetric spaces due to the fact that their structure is much more complicated than that of the underlying group. In recent work, Daniel and Helminck [Daniel, J.R., Helminck, A.G., 2004. Algorithms for computations in local symmetric spaces. Comm. Algebra (in press)] gave a complete set of algorithms for computing the fine structure of Riemannian symmetric spaces. In this paper we make the first step in extending these results to general real reductive symmetric spaces and give a number of algorithms for computing some of their fine structure. This case is a lot more complicated since it involves the intricate relations of five root systems and their Weyl groups instead of just two as in the Riemannian case. We show first that this fine structure can be obtained from the setting of a complex reductive Lie group with a pair of commuting involutions. Then we proceed to give a number of algorithms for computing the fine structure of the latter.