Maximal semigroups in semi-simple Lie groups

The maximal semigroups with nonempty interior in a semi-simple Lie group with finite center are characterized as compression semigroups of subsets in the flag manifolds of the group. For this purpose a convexity theory, called here -convexity, based on the open Bruhat cells is developed. It turns out that a semigroup with nonempty interior is maximal if and only if it is the compression semigroup of the interior of a -convex set.

     

Homogenous projective factors for actions of semi-simple Lie groups

We analyze the structure of a continuous (or Borel) action of a connected semi-simple Lie group G with finite center and real rank at least 2 on a compact metric (or Borel) space X, using the existence of a stationary measure as the basic tool. The main result has the following corollary: Let P be a minimal parabolic subgroup of G, and K a maximal compact subgroup. Let λ be a P-invariant probability measure on X, and assume the P-action on (X,λ) is mixing. Then either λ is invariant under G, or there exists a proper parabolic subgroup Q⊂G, and a measurable G-equivariant factor map ϕ:(X,ν)→(G/Q,m), where ν=∫ _K kλdk and m is the K-invariant measure on G/Q. Furthermore, The extension has relatively G-invariant measure, namely (X,ν) is induced from a (mixing) probability measure preserving action of Q. Oblatum 14-X-1997 & 18-XI-1998 / Published online: 20 August 1999     

Nonreversibility of subsemigroups of semi-simple Lie groups

LetG be a connected semi-simple Lie group withfinite centre andS∋G a subsemigroup withnonvoid interior. The purpose of this article is to show thatS innot simultaneously leftand right reversible unlessS=G.     

Transitive partial actions of groups

J. Kellendonk and M. V. Lawson established that each partial action of a group G on a set Y can be extended to a global action of G on a set Y _G containing a copy of Y. In this paper we classify transitive partial group actions. When G is a topological group acting on a topological space Y partially and transitively we give a condition for having a Hausdorff topology on Y _G such that the global group action of G on Y _G is continuous and the injection Y into Y _G is an open dense equivariant embedding.      

An analogue of Hardy's theorem for semi-simple Lie groups

A well known theorem of Hardy on Fourier transform pairs says that a function on and its Fourier transform cannot both be ``very rapidly decreasing'. We prove here an analogue of this result in the case of semi-simple Lie groups.

     

Bol loops as sections in semi-simple Lie groups of small dimension

Using the relations between the theory of differentiable Bol loops and the theory of affine symmetric spaces we classify all connected differentiable Bol loops having an at most nine-dimensional semi-simple Lie group as the group topologically generated by their left translations. We show that all these Bol loops are isotopic to direct products of Bruck loops of hyperbolic type or to Scheerer extensions of Lie groups by Bruck loops of hyperbolic type.This paper was supported by DAAD.     

Partial actions of groups and actions of inverse semigroups

Given a group , we construct, in a canonical way, an inverse semigroup associated to . The actions of are shown to be in one-to-one correspondence with the partial actions of , both in the case of actions on a set, and that of actions as operators on a Hilbert space. In other words, and have the same representation theory. We show that governs the subsemigroup of all closed linear subspaces of a -graded -algebra, generated by the grading subspaces. In the special case of finite groups, the maximum number of such subspaces is computed. A ``partial' version of the group -algebra of a discrete group is introduced. While the usual group -algebra of finite commutative groups forgets everything but the order of the group, we show that the partial group -algebra of the two commutative groups of order four, namely and , are not isomorphic.

     

Anosov actions of nilpotent Lie groups

We study Anosov actions of nilpotent Lie groups on closed manifolds. Our main result is a generalization to the nilpotent case of a classical theorem by J.F. Plante in the 70's. More precisely, we prove that, for what we call a good Anosov action of a nilpotent Lie group on a closed manifold, if the non-wandering set is the entire manifold, then the closure of stable strong leaves coincide with the closure of the strong unstable leaves. This implies the existence of an equivariant fibration of the manifold onto a homogeneous space of the Lie group, having as fibers the closures of the leaves of the strong foliation.     

Smooth actions of Lie groups and Lie algebras on manifolds

Necessary or sufficient conditions are presented for the existence of various types of actions of Lie groups and Lie algebras on manifolds.     

Symplectically asystatic actions of compact Lie groups

In this paper we introduce the notion of symplectically asystatic Hamiltonian action on a Kahler manifold. In the algebraic setting we prove that if a complex linear group G acts on a Kahler manifold in a symplectically asystatic fashion, then the G-orbits are spherical. Finally, we give the complete classification of symplectically asystatic irreducible representations.     

Covering space semigroups and retracts of compact Lie groups

If B is a compact connected Lie group and N a finite central subgroup, let \({f\colon B\to B/N}\) be the associated covering morphism. The mapping cylinder \({{\mathrm{MC}}(f)}\) is a compact monoid which we call a covering space semigroup. A prominent example is the classical Möbius band \({\mathbb{M}^2}\). An (L)-semigroup is a compact n-manifold X with connected boundary B together with a monoid structure on X such that B is a subsemigroup of X. Every covering space semigroup with \({|N|=2}\) is an (L)-semigroup, and every nonorientable (L)-semigroup is a covering space semigroup. Here \({\mathbb{M}^2}\) is a guiding example. In general, a covering space semigroup X is not a manifold but does have a well-defined manifold boundary. The study of covering space semigroups leads to the following Theorem. Let B be a compact connected Lie group with a central circle group as a direct factor. Then there exist infinitely many pairwise nonisomorphic covering space semigroups with boundary B, and each such semigroup is a retract of a compact connected Lie group.     

Uniqueness of solutions to Schrödinger equations on complex semi-simple Lie groups

In this note we study the time-dependent Schrödinger equation on complex semi-simple Lie groups. We show that if the initial data is a bi-invariant function that has sufficient decay and the solution has sufficient decay at another fixed value of time, then the solution has to be identically zero for all time. We also derive Strichartz and decay estimates for the Schrödinger equation. Our methods also extend to the wave equation. On the Heisenberg group we show that the failure to obtain a parametrix for our Schrödinger equation is related to the fact that geodesics project to circles on the contact plane at the identity.     

Distinguished sets of semi-simple Lie algebras

Journal of Algebraic Combinatorics - We call a finite, spanning set of a semi-simple real Lie algebra a distinguished set if it satisfies the following property: The Lie bracket of any two elements...     

Pseudo-differential operators and Markov semigroups on compact Lie groups

We extend the Ruzhansky-Turunen theory of pseudo-differential operators on compact Lie groups into a tool that can be used to investigate group-valued Markov processes in the spirit of the work in Euclidean spaces of N. Jacob and collaborators. Feller semigroups, their generators and resolvents are exhibited as pseudo-differential operators and the symbols of the operators forming the semigroup are expressed in terms of the Fourier transform of the transition kernel. The symbols are explicitly computed for some examples including the Feller processes associated to stochastic flows arising from solutions of stochastic differential equations on the group driven by Lévy processes. We study a family of Lévy-type linear operators on general Lie groups that are pseudo-differential operators when the group is compact and find conditions for them to give rise to symmetric Dirichlet forms.     

Mixing of all orders of Lie groups actions

Summary We show that for Lie groups whose adjoint representation reflects their topology, mixing implies mixing of all orders. In particular we prove that mixing actions of a semisimple Lie group are mixing of all orders, answering a conjecture of B. Marcus.Oblatum 6-VIII-1990 & 4-IV-1991Sponsored in part by the Edmund Landau Center for research in Mathematical Analysis supported by the Minerva Foundation (Federal Republic of Germany)