Action Of Lie Group Research Articles

Invariant Whitney functions

Theorem 1 of [G. W. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975) 63–68.] says that for a linear action of a compact Lie group [Formula: see text] on a finite dimensional real vector space [Formula: see text], any smooth [Formula: see text]-invariant function on [Formula: see text] can be written as a composite with the Hilbert map. We prove a similar statement for the case of Whitney functions along a subanalytic set [Formula: see text] fulfilling some regularity assumptions. In order to deal with the case when [Formula: see text] is not [Formula: see text]-stable, we use the language of groupoids.

Variable step size commutator free Lie group integrators

We introduce variable step size commutator free Lie group integrators, where the error control is achieved using embedded Runge–Kutta pairs. These are schemes for the integration of initial value problems posed on homogeneous spaces by means of Lie group actions. The focus is on commutator free methods, in which the approximation evolves by composing flows generated by Lie group exponentials. Such methods are encoded by a generalization of Butcher’s Runge–Kutta tableaux, but it is known that more order conditions must be satisfied to obtain a scheme of a given order than are required for classical RK schemes. These extra considerations complicate the task of designing embedded pairs. Moreover, whilst the computational cost of RK schemes is typically dominated by function evaluations, in most situations, the dominant cost of commutator free Lie group integrators comes from computing Lie group exponentials. We therefore give Butcher tableaux for several families of methods of order 3(2) and 4(3), designed with a view to minimizing the number of Lie group exponentials required at each time step, and briefly discuss practical error control mechanisms. The methods are then applied to a selection of examples illustrating the expected behaviour.

Using Lie group integrators to solve two and higher dimensional variational problems with symmetry

The theory of moving frames has been used successfully to solve one dimensional (1D) variational problems invariant under a Lie group symmetry. In the one dimensional case, Noether's laws give first integrals of the Euler–Lagrange equations. In higher dimensional problems, the conservation laws do not enable the exact integration of the Euler–Lagrange system. In this paper we use the theory of moving frames to help solve, numerically, some higher dimensional variational problems, which are invariant under a Lie group action. In order to find a solution to the variational problem, we need first to solve the Euler Lagrange equations for the relevant differential invariants, and then solve a system of linear, first order, compatible, coupled partial differential equations for a moving frame, evolving on the Lie group. We demonstrate that Lie group integrators may be used in this context. We show first that the Magnus expansions on which one dimensional Lie group integrators are based, may be taken sequentially in a well defined way, at least to order 5; that is, the exact result is independent of the order of integration. We then show that efficient implementations of these integrators give a numerical solution of the equations for the frame, which is independent of the order of integration, to high order, in a range of examples. Our running example is a variational problem invariant under a linear action of \begin{document}$ SU(2) $\end{document} . We then consider variational problems for evolving curves which are invariant under the projective action of \begin{document}$ SL(2) $\end{document} and finally the standard affine action of \begin{document}$ SE(2) $\end{document} .

FINITE FLAT SPACES

We say that a finite metric space $X$ can be embedded almost isometrically into a class of metric spaces $C$, if for every $\epsilon > 0$ there exists an embedding of $X$ into one of the elements of $C$ with the bi-Lipschitz distortion less then $1 + \epsilon$. We show that almost isometric embeddability conditions are equal for following classes of spaces (a) Quotients of Euclidean spaces by isometric actions of finite groups, (b) $L_2$-Wasserstein spaces over Euclidean spaces, (c) Compact flat manifolds, (d) Compact flat orbifolds, (e) Quotients of connected compact bi-invariant Lie groups by isometric actions of compact Lie groups. (This one is the most surprising.) We call spaces which satisfy this conditions finite flat spaces. The question about synthetic definition naturally arises. Since Markov type constants depend only on finite subsets we can conclude that connected compact bi-invariant Lie groups and their quotients have Markov type $2$ with constant $1$.

Invariant Markov processes under actions of Lie groups

We present a representation theoryfor invariant Markov processes under Lie group actions, in the spirit of the classic levy-Khinchin representation. We will study invariant Markov processes at three different levels of generality.First, we consider Markov processes in Lie groups that are invariant under translations, and then Markov processes in manifolds that are invariant under transitive group actions.These two types of processes are called levy processes, and they possess a triple representation. The third type of processes are Markov processes that are invariant under non-transitive group actions. Under certain conditions, such a process may be decomposed into a radial part, that is transversal to group orbits, and an angular part, that is along an orbit.The latter is a time inhomogeneous invariant Markov process, and may be represented by a time-dependent triple.

FACTORISATION OF EQUIVARIANT SPECTRAL TRIPLES IN UNBOUNDED -THEORY

We provide sufficient conditions to factorise an equivariant spectral triple as a Kasparov product of unbounded classes constructed from the group action on the algebra and from the fixed point spectral triple. We show that if factorisation occurs, then the equivariant index of the spectral triple vanishes. Our results are for the action of compact abelian Lie groups, and we demonstrate them with examples from manifolds and$\unicode[STIX]{x1D703}$-deformations. In particular, we show that equivariant Dirac-type spectral triples on the total space of a torus principal bundle always factorise. Combining this with our index result yields a special case of the Atiyah–Hirzebruch theorem. We also present an example that shows what goes wrong in the absence of our sufficient conditions (and how we get around it for this example).

On geodesic flows with symmetries and closed magnetic geodesics on orbifolds

Let $Q$ be a closed manifold admitting a locally free action of a compact Lie group $G$. In this paper, we study the properties of geodesic flows on $Q$ given by suitable G-invariant Riemannian metrics. In particular, we will be interested in the existence of geodesics that are closed up to the action of some element in the group $G$, since they project to closed magnetic geodesics on the quotient orbifold $Q/G$.

Generalized Flow-Box property for singular foliations

We introduce a notion of generalized Flow-Box property valid for general singular distributions and sub-varieties (based on a dynamical interpretation). Just as in the usual Flow-Box Theorem, we characterize geometrical and algebraic conditions of (quasi) transversality in order for an analytic sub-variety X (not necessarily regular) to be a section of a line foliation. We also discuss the case of more general foliations. This study is originally motivated by a question of Jean-Francois Mattei (Invent Math 103(2):297–325, 1991, Theorem 3.2.1) about the existence of local slices for a (non-compact) Lie group action.

Hamiltonian actions of unipotent groups on compact K\"ahler manifolds

We study meromorphic actions of unipotent complex Lie groups on compact K\"ahler manifolds using moment map techniques. We introduce natural stability conditions and show that sets of semistable points are Zariski-open and admit geometric quotients that carry compactifiable K\"ahler structures obtained by symplectic reduction. The relation of our complex-analytic theory to the work of Doran--Kirwan regarding the Geometric Invariant Theory of unipotent group actions on projective varieties is discussed in detail.

Twisted equivariant K-theory of compact Lie group actions with maximal rank isotropy

We consider twisted equivariant K--theory for actions of a compact Lie group $G$ on a space $X$ where all the isotropy subgroups are connected and of maximal rank. We show that the associated rational spectral sequence \`a la Segal has a simple $E_2$--term expressible as invariants under the Weyl group of $G$. Namely, if $T$ is a maximal torus of $G$, they are invariants of the $\pi_1(X^T)$-equivariant Bredon cohomology of the universal cover of $X^T$ with suitable coefficients. In the case of the inertia stack $\Lambda Y$ this term can be expressed using the cohomology of $Y^T$ and algebraic invariants associated to the Lie group and the twisting. A number of calculations are provided. In particular, we recover the rational Verlinde algebra when $Y=\{*\}$.

Compact group actions with the Rokhlin property

We provide a systematic and in-depth study of compact group actions with the Rokhlin property. It is show that the Rokhlin property is generic in some cases of interest; the case of totally disconnected groups being the most satisfactory one. One of our main results asserts that the inclusion of the fixed point algebra induces an order-embedding on K-theory, and that it has a splitting whenever it is restricted to finitely generated subgroups. We develop new results in the context of equivariant semiprojectivity to study actions with the Rokhlin property. For example, we characterize when the translation action of a compact group on itself is equivariantly semiprojective. As an application, it is shown that every Rokhlin action of a compact Lie group of dimension at most one is a dual action. Similarly, for an action of a compact Lie group G on C(X), the Rokhlin property is equivalent to freeness together with triviality of the principal G-bundle $X\to X/G$.

Moving frames and conservation laws of a Lagrangian invariant under the Hyperbolic Rotation-Translation group

Noether's First Theorem guarantees conservation laws provided that the Lagrangian is invariant under a Lie group action. In this paper, via the concept of Killing vector fields and the Minkowski metric, we first construct an important Lie group, known as Hyperbolic Rotation-Translation group. Then, according to Gonçalves and Mansfield's method, we obtain the invariantized Euler-Lagrange equations and the space of conservation laws in terms of vectors of invariants and the adjoint representation of a moving frame, for Lagrangians, which are invariant under Hyperbolic Rotation-Translation (or HRT) group action, in the case where the independent variables are not invariant.

Hopf cyclic cohomology and Hodge theory for proper actions on complex manifolds

We introduce two Hopf algebroids associated to a proper and holomorphic Lie group action on a complex manifold. We prove that the cyclic cohomology of each Hopf algebroid is equal to the Dolbeault cohomology of invariant differential forms. When the action is cocompact, we develop a generalized complex Hodge theory for the Dolbeault cohomology of invariant differential forms. We prove that every cyclic cohomology class of these two Hopf algebroids can be represented by a generalized harmonic form. This implies that the space of cyclic cohomology of each Hopf algebroid is finite dimensional. As an application of the techniques developed in this paper, we generalize the Serre duality and prove a Kodaira type vanishing theorem.

Œ[$Q, R$] = 0 and Kostant partition functions

On a polarized compact symplectic manifold endowed with an action of a compact Lie group, in analogy with geometric invariant theory, one can define the space of invariant functions of degree k . A central statement in symplectic geometry, the quantization commutes with reduction hypothesis, is equivalent to saying that the dimension of these invariant functions depends polynomially on k . This statement was proved by Meinrenken and Sjamaar under positivity conditions. In this paper, we give a new proof of this polynomiality property based on a study of the Atiyah–Bott fixed point formula from the point of view of the theory of partition functions, and a technique for localizing positivity.

Certain Types of Complex Lie Group Action

The main aim in this paper is to look for a novel action with new properties on from the , the Literature are concerned with studying the action of of two representations , one is usual and the other is the dual, while our interest in this work is focused on some actions on complex Lie group[10] . Let G be a matrix complex group , and is representation of In this study we will present and analytic the concepts of action of complex group on We recall the definition of tensor product of two representations of group and construct the definition of action of group on , then by using the equivalent relation between and , we get a new action : The two actions are forming smooth representation of This we have new action which called denoted by which acting on 
 This is smooth representation of The theoretical Justifications are developed and prove supported by some concluding remarks and illustrations.

Convexity theorems for the gradient map on probability measures

Abstract Given a Kähler manifold (Z, J, ω) and a compact real submanifold M ⊂ Z, we study the properties of the gradient map associated with the action of a noncompact real reductive Lie group G on the space of probability measures on M. In particular, we prove convexity results for such map when G is Abelian and we investigate how to extend them to the non-Abelian case.

Focal radii of orbits

ABSTRACTWe show that every effective action of a compact Lie group K on a unit sphere admits an explicit orbit whose principal curvatures are bounded from above by .

A unitary ‘quantization commutes with reduction’ map for the adjoint action of a compact Lie group

Let $K$ be a simply connected compact Lie group and $T^{\ast}(K)$ its cotangent bundle. We consider the problem of commutes with reduction for the adjoint action of $K$ on $T^{\ast}(K).$ We quantize both $T^{\ast}(K)$ and the reduced phase space using geometric quantization with half-forms. We then construct a geometrically natural map from the space of invariant elements in the quantization of $T^{\ast}(K)$ to the quantization of the reduced phase space. We show that this map is a constant multiple of a unitary map.

Remarks on Multisymplectic Reduction

The problem of reduction of multisymplectic manifolds by the action of Lie groups is stated and discussed, as a previous step to give a fully covariant scheme of reduction for classical field theories with symmetries.

Lorentzian manifolds with a conformal action of SL(2,R)

We consider conformal actions of simple Lie groups on compact Lorentzian manifolds. Mainly motivated by the Lorentzian version of a conjecture of Lichnerowicz, we establish the alternative: Either the group acts isometrically for some metric in the conformal class, or the manifold is conformally flat – that is, everywhere locally conformally diffeomorphic to Minkowski space-time. When the group is non-compact and not locally isomorphic to SO (1,n), n \geq 2 , we derive global conclusions, extending a theorem of [18] to some simple Lie groups of real-rank 1. This result is also a first step towards a classification of conformal groups of compact Lorentzian manifolds, analogous to a classification of their isometry groups due to Adams, Stuck and, independently, Zeghib [1, 2, 32].