Compact Transformation Groups Research Articles

Circuit complexity for free fermion with a mass quench

By using a recent approach proposed by Hackl et al. to evaluate the complexity of the free fermionic Gaussian state, we compute the complexity of the Dirac vacuum state of the Fermi system with a mass quench. First of all, we review the counting method given by Hackl et al., and demonstrate that the result can be adapted to all of the compact transformation group G. Then, we utilize this result to study the time evolution of the complexity of these states. We show that, for the rotational invariant reference state, the total complexity of the incoming vacuum state will saturate the value of the instantaneous vacuum state at the late times, with a typical timescale to achieve the final stable state. Moreover, we find that the complexity growth under the sudden quench is directly proportional to the mass difference. And all of the results show some similar behaviors with the circuit complexity of Thermofield double states.

Characterizing slices for proper actions of locally compact groups

In his seminal work [13], R. Palais extended a substantial part of the theory of compact transformation groups to the case of proper actions of locally compact groups. Here we extend to proper actions some other important results well known for compact group actions. In particular, we prove that if H is a compact subgroup of a locally compact group G and S is a small (in the sense of Palais) H-slice in a proper G-space, then the action map G×S→G(S) is open. This is applied to prove that the slicing map fS:G(S)→G/H is continuous and open, which provides an external characterization of a slice. Also an equivariant extension theorem is proved for proper actions. As an application, we give a short proof of the compactness of the Banach–Mazur compacta.

Methods for improving estimators of truncated circular parameters

In decision theoretic estimation of parameters in Euclidean space $\mathbb{R}^p$, the action space is chosen to be the convex closure of the estimand space. In this paper, the concept has been extended to the estimation of circular parameters of distributions having support as a circle, torus or cylinder. As directional distributions are of curved nature, existing methods for distributions with parameters taking values in $\mathbb{R}^p$ are not immediately applicable here. A circle is the simplest one-dimensional Riemannian manifold. We employ concepts of convexity, projection, etc., on manifolds to develop sufficient conditions for inadmissibility of estimators for circular parameters. Further invariance under a compact group of transformations is introduced in the estimation problem and a complete class theorem for equivariant estimators is derived. This extends the results of Moors [J. Amer. Statist. Assoc. 76 (1981) 910-915] on $\mathbb{R}^p$ to circles. The findings are of special interest to the case when a circular parameter is truncated. The results are implemented to a wide range of directional distributions to obtain improved estimators of circular parameters.

A Note on Invariant Temporal Functions

The purpose of this article is to present a result on the existence of Cauchy temporal functions invariant by the action of a compact group of conformal transformations in arbitrary globally hyperbolic manifolds. Moreover, the previous results about the existence of Cauchy temporal functions with additional properties on arbitrary globally hyperbolic manifolds are unified in a fully general theorem. The article is written as self-contained as possible, no prior knowledge on Lorentzian geometry is assumed.

Invariant domains of holomorphy: Twenty years later

This review is devoted to the domains of holomorphy invariant under holomorphic actions of real Lie groups. We have collected here the results on this subject obtained during the last twenty years, which have passed since the publication of the first review of the authors on this topic. This first review was mainly devoted to the case of compact transformation groups, while the first two sections of the present review deal mostly with noncompact groups. In Section 3 we discuss the problem of rigidity of automorphism groups of domains of holomorphy invariant under compact transformation groups.

The covering homotopy extension problem for compact transformation groups

It is shown that the orbit space of universal (in the sense of Palais) G-spaces classifies G-spaces. Theorems on the extension of covering homotopy for G-spaces and on a homotopy representation of the isovariant category ISOV are proved.

Retracts and extensors of compact transformation groups

In this paper, we study G-spaces, i.e., topological spaces with actions of a compact group G, from the point of view of the theory of retracts. We recall definitions and properties of absolute retracts, absolute neighborhood retracts, absolute extensors, and absolute neighborhood extensors with respect to the equivariant weakly hereditary class \( \mathfrak{G} \) of G-spaces. Also, we give basic notions of the theory of topological transformation groups.

Shape theory of compact transformation groups

In this paper, we present a systematic investigation of shape theory of compact transformation groups, the so-called equivariant shape theory.

On Murayamaʼs theorem on extensor properties of G-spaces of given orbit types

We develop a method of extending actions of compact transformation groups which is then applied to the problem of preservation of equivariant extensor property by passing to a subspace of given orbit types.

Locally compact (2, 2)-transformation groups

We determine all locally compact imprimitive transformation groups acting sharply 2-transitively on a non-totally disconnected quotient space of blocks inducing on any block a sharply 2-transitive group and satisfying the following condition: if Δ1, Δ2 are two distinct blocks and Pi, Qi ∈ Δi (i = 1, 2), then there is just one element in the inertia subgroup which maps Pi onto Qi. These groups are natural generalizations of the group of affine mappings of the line over the algebra of dual numbers over the field of real or complex numbers or over the skew-field of quaternions. For imprimitive locally compact groups, our results correspond to the classical results of Kalscheuer for primitive locally compact groups (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

On extending actions of groups

Problems of dense and closed extension of actions of compact transformation groups are solved. The method developed in the paper is applied to problems of extension of equivariant maps and of construction of equivariant compactifications.Bibliography: 27 titles.

Flat bases of invariant polynomials and P̂-matrices of E7 and E8

Let G be a compact group of linear transformations of a Euclidean space V. The G-invariant C∞ functions can be expressed as C∞ functions of a finite basic set of G-invariant homogeneous polynomials, sometimes called an integrity basis. The mathematical description of the orbit space V/G depends on the integrity basis too: it is realized through polynomial equations and inequalities expressing rank and positive semidefiniteness conditions of the P̂-matrix, a real symmetric matrix determined by the integrity basis. The choice of the basic set of G-invariant homogeneous polynomials forming an integrity basis is not unique, so it is not unique the mathematical description of the orbit space too. If G is an irreducible finite reflection group, Saito et al. [Commun. Algebra 8, 373 (1980)] characterized some special basic sets of G-invariant homogeneous polynomials that they called flat. They also found explicitly the flat basic sets of invariant homogeneous polynomials of all the irreducible finite reflection groups except of the two largest groups E7 and E8. In this paper the flat basic sets of invariant homogeneous polynomials of E7 and E8 and the corresponding P̂-matrices are determined explicitly. Using the results here reported one is able to determine easily the P̂-matrices corresponding to any other integrity basis of E7 or E8. From the P̂-matrices one may then write down the equations and inequalities defining the orbit spaces of E7 and E8 relatively to a flat basis or to any other integrity basis. The results here obtained may be employed concretely to study analytically the symmetry breaking in all theories where the symmetry group is one of the finite reflection groups E7 and E8 or one of the Lie groups E7 and E8 in their adjoint representations.

Classification of compact transformation groups on complex quadrics with codimension one orbits

Let $G$ be a compact connected Lie group and $M$ a rational cohomology complex quadric of real dimension divisible by $4$ (where $\dim M\neq 4$). The aim of this paper is to classify pairs $(G,M)$ such that $G$ acts smoothly on $M$ with codimension one principal orbits. There exist eight such pairs up to essential isomorphism. The underlying manifold $M$ is diffeomorphic to the genuine complex quadric except one pair.

16-dimensional compact projective planes with 3 fixed points

Let 9 = (P, £) be a topological projective plane with a compact point set P of finite (covering) dimension d = dim P > 0. A systematic treatment of such planes can be found in the book Compact Projective Planes [15]. Each line L e £ is homotopy equivalent to a sphere §/ with / 1 8, and d = 2/, see [15] (54.1 1). In all known examples, L is in fact homeomorphic to §/. Taken with the compact-open topology, the automorphism group Σ = Aut^ (of all continuous collineations) is a locally compact transformation group of Ρ with a countable basis, the dimension dim Σ is finite [15] (44.3 and 83.2). The classical examples are the planes κ over the three locally compact, connected fields IK with ( — dim IK and the 16-dimensional Moufang plane G = έΡ® over the octonion algebra Θ. If 9 is a classical plane, then Aut^ is an almost simple Lie group of dimension Q, where C = 8, €2 = 16, €4 = 35, and Cg = 78. In all other cases, dim Σ < ^ Q + 1 < 5Λ Planes with a group of dimension sufficiently close to C( can be described explicitly. More precisely, the classification program seeks to determine all pair s (&, Δ), where Δ is a connected closed subgroup of Aut & and bf ^ dim Δ 5^ for a suitable bound be 4/ — 1 . This has been accomplished for t ^ 2 and also for 64 = 17. Here, the case ( = 8 will be studied; the value of bt varies with the configuration of the fixed elements ofA. Most theorems that have been obtained so far require additional assumptions on the structure of Δ. If dim Δ 27, then Δ is always a Lie group [12]. By the structure theory of Lie groups, there are 3 possibilities: (i) Δ is semi-simple, or (ii) Δ contains a central torus subgroup, or (iii) Δ has a minimal normal vector subgroup, cf. [15] (94.26). The first two cases are understood fairly well:

How entangled pairs act as measuring rods on manifolds of generalized coherent states

Generalized coherent states arise from reference states by the action of locally compact transformation groups and thereby form manifolds on which there is an invariant measure. It is shown that this implies the existence of canonically associated Bell states that serve as measuring rods by relating the metric geometry of the manifold to the observed EPR correlations. It is further shown that these correlations can be accounted for by a hidden variable theory which is non-local but invariant under the stability group of the reference state.

Integrable actions and transformation groups whose C*-algebras have bounded trace

Let (G,X) be a locally compact transformation group. We characterize when the associated transformation-group C*-algebra C 0 (X) x G has bounded trace. If G acts freely on X, then, C 0 (X) x G has bounded trace if and only if the action of G on C 0 (x) is integrable. If G is abelian and the stability subgroups vary continuously, then C 0 (X) x G has bounded trace if and only if the action is integrable relative to the stability subgroups. The upper multiplicity of an irreducible representation of C 0 (X) x G is invariant under the dual action of G; as an application we identify the largest bounded-trace ideal in Type I transformation-group C*-algebras.

The Transformation Groups Whose C *-Algebras are Fell Algebras

Let ( G , X ) be a locally compact transformation group in which G acts freely on X . We show that the associated transformation-group C *-algebra C 0 ( X ) [rtimes ] G is a Fell algebra if and only if X is a Cartan G -space.

An equivariant Brauer semigroup and the symmetric imprimitivity theorem

Suppose that ( X , G ) (X,G) is a second countable locally compact transformation group. We let S G ⁡ ( X ) \operatorname {S}_G(X) denote the set of Morita equivalence classes of separable dynamical systems ( A , G , α ) (A,G,\alpha ) where A A is a C 0 ( X ) C_{0}(X) -algebra and α \alpha is compatible with the given G G -action on X X . We prove that S G ⁡ ( X ) \operatorname {S}_{G}(X) is a commutative semigroup with identity with respect to the binary operation [ A , G , α ] [ B , G , β ] = [ A ⊗ X B , G , α ⊗ X β ] [A,G,\alpha ][B,G,\beta ]=[A\otimes _{X}B,G,\alpha \otimes _{X}\beta ] for an appropriately defined balanced tensor product on C 0 ( X ) C_{0}(X) -algebras. If G G and H H act freely and properly on the left and right of a space X X , then we prove that S G ⁡ ( X / H ) \operatorname {S}_{G}(X/H) and S H ⁡ ( G ∖ X ) \operatorname {S}_{H}(G\setminus X) are isomorphic as semigroups. If the isomorphism maps the class of ( A , G , α ) (A,G,\alpha ) to the class of ( B , H , β ) (B,H,\beta ) , then A ⋊ α G A\rtimes _{\alpha }G is Morita equivalent to B ⋊ β H B\rtimes _{\beta }H .

Global G-Manifold Reductions and Resolutions

The purpose of this note is to exhibit some simple and basic constructions for smooth compact transformation groups, and some of their most immediate applications to geometry.

A global slice theorem for proper Hamiltonian actions

An analogue of a theorem of Abels [A] reducing the proper action of a non-compact Lie group to a compact transformation group is proved for the Hamiltonian setting.