Actions Of Compact Lie Groups Research Articles

Homotopy of planar Lie group equivariant presheaves

We utilise the theory of crossed simplicial groups to introduce a collection of local Quillen model structures on the category of simplicial presheaves with a compact planar Lie group action on a small Grothendieck site. As an application, we give a characterisation of equivariant cohomology theories on a site as derived mapping spaces in these model categories.

Hamiltonian group actions on exact symplectic manifolds with proper momentum maps are standard

We give a complete characterization of Hamiltonian actions of compact Lie groups on exact symplectic manifolds with proper momentum maps. We deduce that every Hamiltonian action of a compact Lie group on a contractible symplectic manifold with a proper momentum map is globally linearizable.

Hamiltonian and symplectic symmetries: An introduction

Classical mechanical systems are modeled by a symplectic manifold $(M,\omega)$, and their symmetries, encoded in the action of a Lie group $G$ on $M$ by diffeomorphisms that preserves $\omega$. These actions, which are called symplectic, have been studied in the past forty years, following the works of Atiyah, Delzant, Duistermaat, Guillemin, Heckman, Kostant, Souriau, and Sternberg in the 1970s and 1980s on symplectic actions of compact abelian Lie groups that are, in addition, of type, i.e. they also satisfy Hamilton's equations. Since then a number of connections with combinatorics, finite dimensional integrable Hamiltonian systems, more general symplectic actions, and topology, have flourished. In this paper we review classical and recent results on Hamiltonian and non Hamiltonian symplectic group actions roughly starting from the results of these authors. The paper also serves as a quick introduction to the basics of symplectic geometry.

The equivariant Cuntz semigroup

We introduce an equivariant version of the Cuntz semigroup that takes an action of a compact group into account. The equivariant Cuntz semigroup is naturally a semimodule over the representation semiring of the given group. Moreover, this semimodule satisfies a number of additional structural properties. We show that the equivariant Cuntz semigroup, as a functor, is continuous and stable. Moreover, cocycle conjugate actions have isomorphic associated equivariant Cuntz semigroups. One of our main results is an analog of Julg's theorem: the equivariant Cuntz semigroup is canonically isomorphic to the Cuntz semigroup of the crossed product. We compute the induced semimodule structure on the crossed product, which in the abelian case is given by the dual action. As an application of our results, we show that freeness of a compact Lie group action on a compact Hausdorff space can be characterized in terms of a canonically defined map into the equivariant Cuntz semigroup, extending results of Atiyah and Segal for equivariant K-theory. Finally, we use the equivariant Cuntz semigroup to classify locally representable actions on direct limits of one-dimensional NCCW-complexes, generalizing work of Handelman and Rossmann.

Rokhlin dimension for compact group actions

We introduce and systematically study the notion of Rokhlin dimension (with and without commuting towers) for compact group actions on $C^*$-algebras. This notion generalizes the one introduced by Hirshberg, Winter and Zacharias for finite groups, and contains the Rokhlin property as the zero dimensional case. We show, by means of an example, that commuting towers cannot always be arranged, even in the absence of $K$-theoretic obstructions. For a compact Lie group action on a compact Hausdorff space, freeness is equivalent to finite Rokhlin dimension of the induced action. We compare the notion of finite Rokhlin dimension to other existing definitions of noncommutative freeness for compact group actions. We obtain further $K$-theoretic obstructions to having an action of a non-finite compact Lie group with finite Rokhlin dimension with commuting towers, and use them to confirm a conjecture of Phillips.

The Grauert–Grothendieck complex on differentiable spaces and a sheaf complex of Brylinski

We use the Grauert–Grothendieck complex on differentiable spaces to study basic relative forms on the inertia space of a compact Lie group action on a manifold. We prove that the sheaf complex of basic relative forms on the inertia space is a fine resolution of Bryliski’s sheaf of functions on the inertia space.

Positivity of character subbundles and minimum principle for noncompact group actions

Based on the results of Berndtsson on the positivity of holomorphic vector bundles associated to holomorphic fibration, we prove that, under the setting of the compact Lie group actions, the character subbundles of such bundles are holomorphic vector bundles and positive in the sense of Nakano. The second main purpose of the present paper is to establish Berndtsson’s integral form of Kiselman’s minimum principle in the setting of the noncompact Lie group actions.

Orientation and Symmetries of Alexandrov Spaces with Applications in Positive Curvature

We develop two new tools for use in Alexandrov geometry: a theory of ramified orientable double covers and a particularly useful version of the Slice Theorem for actions of compact Lie groups. These tools are applied to the classification of compact, positively curved Alexandrov spaces with maximal symmetry rank.

Compact Lie group action with the continuous Rokhlin property

In this paper, we study the continuous Rokhlin property of C⁎-dynamical systems using techniques of equivariant KK-theory and quantum group theory. In particular, we determine the KK-equivalence class and give a classification of Kirchberg G-algebras when the G is a compact Lie group with the Hodgkin condition.

Classifying spaces of twisted loop groups

We study the classifying space of a twisted loop group $L_{\sigma}G$ where $G$ is a compact Lie group and $\sigma$ is an automorphism of $G$ of finite order modulo inner automorphisms. Equivalently, we study the $\sigma$-twisted adjoint action of $G$ on itself. We derive a formula for the cohomology ring $H^*(BL_{\sigma}G)$ and explicitly carry out the calculation for all automorphisms of simple Lie groups. More generally, we derive a formula for the equivariant cohomology of compact Lie group actions with constant rank stabilizers.

Higher-order orbifold Euler characteristics for compact Lie group actions

We generalize the notions of the orbifold Euler characteristic and of the higher-order orbifold Euler characteristics to spaces with actions of a compact Lie group using integration with respect to the Euler characteristic instead of the summation over finite sets. We show that the equation for the generating series of the kth-order orbifold Euler characteristics of the Cartesian products of the space with the wreath products actions proved by Tamanoi for finite group actions and by Farsi and Seaton for compact Lie group actions with finite isotropy subgroups holds in this case as well.

Weyl group symmetry on the GKM graph of a GKM manifold with an extended Lie group action

We consider the class of manifolds with compact Lie group actions which restrict to GKM-actions on the maximal torus. First, we see their GKM-graphs admit symmetry of the Weyl groups. And then, we study its combinatorial abstraction; starting with abstract GKM-graphs with symmetry, we derive certain properties which reflect topology in a purely combinatorial way.

Stratifications of Inertia Spaces of Compact Lie Group Actions

We study the topology of the inertia space of a smooth $G$-manifold $M$ where $G$ is a compact Lie group. We construct an explicit Whitney stratification of the inertia space, demonstrating that the inertia space is a triangulable differentiable stratified space. In addition, we demonstrate a de Rham theorem for differential forms defined on the inertia space with respect to this stratification.

K-theory and the quantization commutes with reduction problem

The authors examine the quantization commutes with reduction phenomenon for Hamiltonian actions of compact Lie groups on closed symplectic manifolds from the point of view of topological K-theory and K-homology. They develop the machinery of K-theory wrong-way maps in the context of orbifolds and use it to relate the quantization commutes with reduction phenomenon to Bott periodicity and the K-theory formulation of the Weyl character formula.

Genericity in equivariant dynamical systems and equivariant Fuller index theory

We define a notion of equivariant non-degeneracy of G-maps to introduce the class of equivariantly non-degenerate flows on smooth compact manifolds with compact Lie group action. We prove genericity of this class and use this result to construct an equivariant version of the Fuller index, which detects group orbits of periodic orbits of the flow, distinguished by their isotropy.

Quantizations of compact Lie group actions

In this paper we describe quantizations in the monoidal categories of unitary representations of compact connected Lie groups. For the n-dimensional torus T we show that the set Q(T) of quantizations is isomorphic to the (n2)-dimensional torus. For connected compact Lie groups G of rank n, we get the result that the set QE(G) of extendible quantizations of G-modules is isomorphic to the set of quantizations of its maximal torus T invariant under action by its Weyl group. For all these cases we give explicit formulae for quantizations and apply these to quantize Hilbert–Schmidt operators.

Morse–Bott inequalities in the presence of a compact Lie group action and applications

In this paper, we obtain Morse–Bott inequalities in the presence of a compact Lie group action via Bismut–Lebeauʼs analytic localization techniques. As an application, we obtain Morse–Bott inequalities on compact manifold with nonempty boundary by applying the generalized Morse–Bott inequalities to the doubling manifold.

K-theory of equivariant quantization

Using an equivariant version of Connesʼ Thom isomorphism, we prove that equivariant K-theory is invariant under strict deformation quantization for a compact Lie group action.

Coupling symmetries with Poisson structures

We study local normal forms for completely integrable systems on Poisson manifolds in the presence of additional symmetries. The symmetries that we consider are encoded in actions of compact Lie groups. The existence of Weinstein’s splitting theorem for the integrable system is also studied giving some examples in which such a splitting does not exist, i.e. when the integrable system is not, locally, a product of an integrable system on the symplectic leaf and an integrable system on a transversal. The problem of splitting for integrable systems with additional symmetries is also considered.

Non-symplectic actions on complex projective spaces

We construct smooth actions of arbitrary compact Lie groups on complex projective spaces, such that the corresponding transformations arising from the group action do not preserve any symplectic structure on the complex projective space.