Actions Of Compact Lie Groups Research Articles

A note on equivariant normal forms of Poisson structures

We prove an equivariant version of the local splitting theorem for tame Poisson structures and Poisson actions of compact Lie groups. As a consequence, we obtain an equivariant linearization result for Poisson structures whose transverse structure has semisimple linear part of compact type.

On the orders of finite semisimple groups

The aim of this paper is to investigate the order coincidences among the finite semisimple groups and to give a reasoning of such order coincidences through the transitive actions of compact Lie groups.

Complete involutive algebras of functions on cotangent bundles of homogeneous spaces

Homogeneous spaces of all compact Lie groups admit Riemannian metrics with completely integrable geodesic flows by means of C ∞ –smooth integrals [9, 10]. The purpose of this paper is to give some constructions of complete involutive algebras of analytic functions, polynomial in velocities, on the (co)tangent bundles of homogeneous spaces of compact Lie groups. This allows us to obtain new integrable Riemannian and sub-Riemannian geodesic flows on various homogeneous spaces, such as Stiefel manifolds, flag manifolds and orbits of the adjoint actions of compact Lie groups.

Copolarity of isometric actions

We introduce a new integral invariant for isometric actions of compact Lie groups, the copolarity. Roughly speaking, it measures how far from being polar the action is. We generalize some results about polar actions in this context. In particular, we develop some of the structural theory of copolarity k representations, we classify the irreducible representations of copolarity one, and we relate the copolarity of an isometric action to the concept of variational completeness in the sense of Bott and Sainelson.

Borsuk-Ulam type theorems for compact Lie group actions

Borsuk-Ulam type theorems for arbitrary compact Lie group actions are proven. The transfer plays a major role in this approach.

Proper actions on cohomology manifolds

Essential results about actions of compact Lie groups on connected manifolds are generalized to proper actions of arbitrary groups on connected cohomology manifolds. Slices are replaced by certain fiber bundle structures on orbit neighborhoods. The group dimension is shown to be effectively finite. The orbits of maximal dimension form a dense open connected subset. If some orbit has codimension at most 2, then the group is effectively a Lie group.

A higher Lefschetz formula for flat bundles

In this paper, we prove a fixed point formula for flat bundles. To this end, we use cyclic cocycles which are constructed out of closed invariant currents. We show that such cyclic cocycles are equivariant with respect to isometric longitudinal actions of compact Lie groups. This enables us to prove fixed point formulae in the cyclic homology of the smooth convolution algebra of the foliation.

Homogeneity Rank and Atoms of Actions

We introduce two related concepts for smooth actions of compact Lie groups:The homogeneity rank is a simple numerical invariant of the action.As one of our results we determine the precise range of this invariantfor isometric actions on compact Riemannian manifolds with positivesectional curvature and exhibit special properties of the actionswith maximal homogeneity rank.Atoms are special components of fixed point sets. They inherit actionswith the same cohomogenity and homogeneity rank as the original action,but with trivial principal isotropy group. Other properties of the originalaction like polarity are reflected in the atoms.We determine the atoms in some interesting concrete cases.Not only for this purpose we give a detailed treatise on the structureof fixed point sets, in particular in cohomogeneity one manifolds.

Classification of equivariant real vector bundles over a circle

This is a continuation of the previous work [CKMS01] by the authors on classification of equivariant complex vector bundles over a circle. In this paper equivariant real vector bundles over a circle with a compact Lie group action are classified, by characterizing their fiber representations, and by using the result of the complex case. Their triviality is also treated. The basic phenomenon is similar to the complex case but more complicated here.

Localization of the Riemann–Roch Character

We present a K-theoretic approach to the Guillemin–Sternberg conjecture (V. Guillemin and S. Sternberg, Invent. Math.67 (1982), 515–538), about the commutativity of geometric quantization and symplectic reduction, which was proved by E. Meinrenken (J. Amer. Math. Soc.9 (1996), 373–389; Adv. Math.134, (1998), 240–277) and Tian–Zhang (Y. Tian and W. Zhang, Invent. Math.132 (1998), 229–259). Besides providing a new proof of this conjecture for the full non-Abelian group action case, our methods lead to a generalization for compact Lie group actions on manifolds that are not symplectic; these manifolds carry an invariant almost complex structure and an abstract moment map.

Compact Lie groups acting on pseudomanifolds

In this paper we introduce the concept of a $G$-pseudomanifold, which is an equivariant version of the stratified spaces defined by Goresky and Mac Pherson for compact Lie group actions.

BRST Cohomology and Phase Space Reduction in Deformation Quantization

In this article we consider quantum phase space reduction when zero is a regular value of the momentum map. By analogy with the classical case we define the BRST cohomology in the framework of deformation quantization. We compute the quantum BRST cohomology in terms of a `quantum' Chevalley-Eilenberg cohomology of the Lie algebra on the constraint surface. To prove this result, we construct an explicit chain homotopy, both in the classical and quantum case, which is constructed out of a prolongation of functions on the constraint surface. We have observed the phenomenon that the quantum BRST cohomology cannot always be used for quantum reduction, because generally its zero part is no longer a deformation of the space of all smooth functions on the reduced phase space. But in case the group action is `sufficiently nice', e.g. proper (which is the case for all compact Lie group actions), it is shown for a strongly invariant star product that the BRST procedure always induces a star product on the reduced phase space in a rather explicit and natural way. Simple examples and counter examples are discussed.

On finitistic spaces

Finitistic spaces form a natural class containing compact and finite-dimensional spaces. Introduced and investigated by fixed-point theorists, finitistic spaces found an application in cohomological dimension theory. In the paper, two characterizations of paracompact, finitistic spaces are proved. These characterizations allow to create a mechanism of generalizing results of finite dimension theory. As an application we obtain results on compact group actions on paracompact spaces which were previously known for compact Lie group actions.

Ergodic Actions of Universal Quantum Groups on Operator Algebras

We construct ergodic actions of compact quantum groups on C^*-algebras and von Neumann algebras, and exhibit phenomena of such actions that are of a different nature from ergodic actions of compact Lie groups. In particular, we construct: (1). ergodic actions of the compact quantum groups $A_u(Q)$ on the Powers factors; (2). ergodic actions of the compact quantum groups $A_u(n)$ on the hyperfinite II_1 factor; (3). ergodic actions of the compact quantum groups $A_u(Q)$ on the Cuntz algebras; (4). ergodic actions of general compact quantum groups on their homogeneous spaces and an example of a non-homogeneous classical space that nevertheless admits an ergodic action of a compact quantum group.

On Poisson actions of compact Lie groups on symplectic manifolds

Let G(P) be a compact simple Poisson-Lie group equipped with a Poisson structure P, and (M, omega) be a symplectic manifold. Assume that M carries a Poisson action of G(P), and there is an equivariant moment map in the sense of Lu and Weinstein which maps

A convexity theorem for Poisson actions of compact Lie groups

When a Poisson Lie group acts on a symplectic manifold, there is a momentum mapping to the dual group. We prove that the image of this momentum mapping meets the positive Weyl chamber in the exponential of a convex set. The result adapts the well-known theorem of Guillemin, Stemberg, and Kirwan to a multiplicative (rather than additive) setting.

Morse indices of critical manifolds generated by min‐max methods with compact lie group actions and applications

AbstractWe derive some upper and lower bounds for Morse indices of critical manifolds generated by min‐max principles for functionals invariant under a general compact Lie group or a finite group action. The results generalize the similar results in the nonequivariant (no group action) case. In doing so, we also generalize the extension theorem of Dugundji type in the nonequivariant case to the equivariant (group action) case. As an application, we obtain a precise growth estimate for the whole sequence of critical values given by the min‐max procedure for some superquadratic second‐order differential equations. It is well‐known that this growth estimate is crucial in showing the existence of multiple solutions of some superquadratic perturbed Hamiltonian systems and equations.

ON ORBIT CONNECTEDNESS, ORBIT CONVEXITY, AND ENVELOPES OF HOLOMORPHY

The univalence and description of the envelope of holomorphy is considered for a domain having a compact Lie group action. The main result is the following: Let be a holomorphic Stein -manifold, and a -invariant orbit connected domain. Then is schlicht and orbit convex if and only if is schlicht. Moreover, in this case, .

Equivariant Analytic Torsion for Compact Lie Group Actions

We define the equivariant analytic torsion for a compact Lie group action and study its dependence on the geometric data.

Generalized Rochlin invariants of fixed point sets

The Rochlin invariant for 3-manifolds is extended to higher dimensions using a result of Ochanine. Given a semifree differentiable S-action on a closed mod 2 homology sphere M with fixed point set F , generalized Rochlin invariants are definable for both M and F , and one result of this paper states that these two invariants are equal. This yields restrictions on the types of semifree differentiable S-actions that some homology spheres can support and the fixed point sets of actions on homology (8k + 7)-spheres. An action of a group G on a space X is said to be semifree if for each x ∈ X either X is fixed under every element of G or else x is not fixed by any element of G except the identity. During the nineteen sixties and seventies it became apparent that the techniques of differential topology had numerous applications to differentiable actions of compact Lie groups (cf. [Bro2], [BP], [RS]). In particular, these and previously developed techniques yielded considerable information on semifree differentiable actions of S and S on spheres. One result was a complete description of the homeomorphism types of the possible fixed point sets. Specifically, these are all closed manifolds with the same integral homology as a sphere of some appropriate dimension (see [HH, Ch. V,§4]). On the other hand, questions about the diffeomorphism types of the fixed point sets are more difficult to answer. In this paper we shall prove a result (Theorem B below) that complements previous work on the smooth realization question; this is a special case of a more general result (Theorem A) relating the diffeomorphism type of the fixed point set to the diffeomorphism type of the ambient manifold. Although evidence suggests that an analog of Theorem B holds for semifree S-actions (see Proposition 3.2 and [Sc4]), the proof of such an analog seems likely to require additional input. The proofs of Theorems A and B involve a higher dimensional analog of the well known Rochlin invariant for closed homology 3-spheres (e.g., see [EK], where it is called the μ-invariant).