Margulis lemma for compact lie groups

We improve Margulis lemma for a compact connected Lie group G: there is a neighborhood U of the identity such that for any finite subgroup , generates an abelian group. We show that for each n, there exists an integer , such that if H is a closed subgroup of a compact connected Lie group G of dimension n, then the quotient group, H/H ₀, has an abelian subgroup of index , where H ₀ is the identity component of H. As an application, we show that the fundamental group of the homogeneous space G/H has an abelian subgroup of index . We show this same property for the fundamental groups of almost non-negatively curved n-manifolds whose universal coverings are not collapsed. X. Rong: supported partially by NSF Grant DMS 0504534 and by a reach found from Beijing Normal University. Y. Wang: supported partially by LMAM of Peking University and by NSFC 10671018.     

Locally compact connected groups acting on euclidean space with lie isotropy groups are lie

It is shown that a locally compact connected group must be a Lie group if it acts on a space with certain local and global cohomological properties (for example, euclidean space and even-dimensional spheres) in such a way that all isotropy groups are Lie groups. Apart from the obvious application to free actions, this result can also be applied to automorphism groups of continous incidence geometries of a particular kind (four-dimensional stable planes).     

Polynomially convex orbits of compact lie groups

LetV be a finite dimensional complex linear space and letG be a compact subgroup of GL(V). We prove that an orbitG, V, is polynomially convex if and only ifG is closed andG is the real form ofG . For every orbitG which is not polynomially convex we construct an analytic annulus or strip inG with the boundary inG. It is also proved that the group of holomorphic automorphisms ofG which commute withG acts transitively on the set of polynomially convexG-orbits. Further, an analog of the Kempf-Ness criterion is obtained and homogeneous spaces of compact Lie groups which admit only polynomially convex equivariant embeddings are characterized.Supported by Federal program Integratsiya, no. 586.Supported by INTAS grant 97/10170.     

The homotopy types of compact lie groups

Hans Scheerer proved that if two simply connected compact Lie groups are homotopically equivalent, then the groups are isomorphic. We give a conceptually simpler proof which shows that the result depends only on the 2 and 3 primary homotopy of the Lie groups.     

Rational Mackey functors for compact lie groups I

A Mackey functor M is a structure analogous to the representationring functor H R(H) encoding good formal behaviour under inductionand restriction. More explicitly, M associates an abelian groupM(H) to each closed subgroup H of a fixed compact Lie groupG, and to each inclusion K H it associates a restriction map and an induction map . This paper gives an analysis of thecategory of Mackey functors M whose values are rational vectorspaces: such a Mackey functor may be specified by giving a suitablycontinuous family consisting of a Q ₀(W_G(H))-module V(H) foreach closed subgroup H with restriction maps V(K) V(K) wheneverK is normal in K and K/K is a torus (a ‘continuous Weyl-toralmodule’). We show that the category of rational Mackeyfunctors is equivalent to the category of rational continuousWeyl-toral modules. In Part II this will be used to give analgebraic analysis of the category of rational Mackey functors,showing in particular that it has homological dimension equalto the rank of the group. 1991 Mathematics Subject Classification:19A22, 20C99, 22E15, 55N91, 55P42, 55P91.     

Bogolyubov averaging and normalization procedures in nonlinear mechanics. I

We suggest a new method for asymptotic analysis of nonlinear dynamical systems based on group-the-oretic methods. On the basis of the Bogolyubov averaging method, we develop a new normalization procedure — asymptotic decomposition. We clarify the contribution of this procedure to the interpretation and development of the averaging method for systems in the standard form and systems with several fast variables. According to this method, the centralized system is regarded as a direct analog of the system averaged in Bogolyubov's sense. The operation of averaging is interpreted as the Bogolyubov projector, i.e., the operation of projection of an operator onto the algebra of centralizer.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1171–1188, September, 1994.     

Bogolyubov averaging and normalization procedures in nonlinear mechanics. III

We describe the technique of normalization based on the method of asymptotic decomposition in the space of representation of a finite-dimensional Lie group. The main topics of the theory necessary for understanding the method are outlined. Models based on the Van der Pol equation are investigated by the method of asymptotic decomposition in the space of homogeneous polynomials (the space of representation of a general linear group in a plane) and in the space of representation of a rotation group on a plane (ordinary Fourier series). The comparison made shows a dramatic decrease in the necessary algebraic manipulations in the second case. We also discuss other details of the technique of normalization based on the method of asymptotic decomposition.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 12, pp. 1627–1646, December, 1994.This research was partially supported by the International Scientific Foundation, grant No. UB2000.     

Bogolyubov averaging and normalization procedures in nonlinear mechanics. II

By using a new method suggested in the first part of the present work, we study systems which become linear in the zero approximation and have perturbations in the form of polynomials. This class of systems has numerous applications. The following fact is even more important: Our technique demonstrates how to generalize the classical method of Poincaré-Birkhoff normal forms and obtain new results by using group-theoretic methods. After a short exposition of the general theory of the method of asymptotic decomposition, we illustrate the new normalization technique as applied to models based on the Lotka-Volterra equations.Published in Ukrainskii Matematicheskii Zhurnal, Vol.46, No. 11, pp. 1509–1526, November, 1994.The present work was supported by the Ukrainian State Committee on Science and Technology.     

Bogolyubov averaging and normalization procedures in nonlinear mechanics. IV

In this paper, we apply the theory developed in parts I-III [Ukr. Math. Zh.,46, No. 9, 1171–1188; No. 11, 1509–1526; No. 12, 1627–1646 (1994)] to some classes of problems. We consider linear systems in zero approximation and investigate the problem of invariance of integral manifolds under perturbations. Unlike nonlinear systems, linear ones have centralized systems, which are always decomposable. Moreover, restrictions connected with the impossibility of diagonalization of the coefficient matrix in zero approximation are removed. In conclusion, we apply the method of local asymptotic decomposition to some mechanical problems.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 8, pp. 1044–1068, August, 1995.This research was partially supported by the International Science Foundation, grant No. UB2000.     

On based-free actions of compact lie groups on the hilbert cube

It is proved that a based-free action α of a given compact Lie groupG on the Hilbert cubeQ is equivalent to the standard based-free action σ if and only if the orbit spaceQ ₀/α of the free partQ ₀=Q* is aQ-manifold having the proper homotopy type of the orbit spaceQ ₀/σ. The existence of an equivariant retraction (Q ₀, σ)→(Q ₀, α) is established. It is proved that for any TikhonovG-spaceX the family of all equivariant mapsX→ conG separates the points and the closed sets inX. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 163–174, February, 1999.     

On quotient spaces of compact lie groups by Tori centralizers

We consider the graph of the homogeneous space K/L, where K is a compact Lie group and L is the centralizer of a torus in K. We obtain a characterization of those spaces whose graphs admit embeddings in a certain standard graph. We compute the number of arcs in such graphs. We also give a simple expression for the Euler class of the homogeneous space K/L.     

Geodesics as products of one-parameter subgroups in compact lie groups and homogeneous spaces

We study the geodesic equation for compact Lie groups G and homogeneous spaces $G/H$, and we prove that the geodesics are orbits of products $\exp (t{X}_1)\cdots \exp (t{X}_N)$ of one-parameter subgroups of G, provided that a simple algebraic condition for the Riemannian metric is satisfied. For the group $SO(3)$, we relate this type of geodesics to the free motion of a symmetric top. Moreover, by using series of Lie subgroups of G, we construct a wealth of metrics having the aforementioned type of geodesics.     

A simple proof of the divergence of Fourier series on compact connected semisimple lie groups

We give an elementary proof of the result proved in [5] on theL ^p divergence of Fourier series on compact connected semisimple Lie groups.     

Control theory on lie groups

Lecture notes of an introductory course on control theory on Lie groups. Controllability and optimal control for left-invariant problems on Lie groups are addressed. A general theory is accompanied by concrete examples. The course is intended for graduate students; no preliminary knowledge of control theory or Lie groups is assumed. Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 27, Optimal Control, 2007.