Compact Group Research Articles

Yang–Mills–Stueckelberg theories, framing and local breaking of symmetries

We consider Yang–Mills theory with a compact structure group [Formula: see text] on a Lorentzian 4-manifold [Formula: see text] such that gauge transformations become identity on a submanifold [Formula: see text] of [Formula: see text] (framing over [Formula: see text]). The space [Formula: see text] is not necessarily a boundary of [Formula: see text] and can have dimension [Formula: see text]. Framing of gauge bundles over [Formula: see text] demands introduction of a [Formula: see text]-valued function [Formula: see text] with support on [Formula: see text] and modification of Yang–Mills equations along [Formula: see text]. The fields [Formula: see text] parametrize non-equivalent flat connections mapped into each other by a dynamical group [Formula: see text] changing gauge frames over [Formula: see text]. It is shown that the charged condensate [Formula: see text] is the Stueckelberg field generating an effective mass of gluons in the domain [Formula: see text] of space [Formula: see text] and keeping them massless outside [Formula: see text]. We argue that the local Stueckelberg field [Formula: see text] can be responsible for color confinement. We also briefly discuss local breaking of symmetries in gravity. It is shown that framing of the tangent bundle over a subspace of spacetime makes gravitons massive in this subspace.

Group Actions on Product Systems

We introduce the concept of a crossed product of a product system by a locally compact group. We prove that the crossed product of a row-finite and faithful product system by an amenable group is also a row-finite and faithful product system. We generalize a theorem of Hao and Ng about the crossed product of the Cuntz-Pimsner algebra of a $C^{\ast}$-correspondence by a group action to the context of product systems. We present examples related to group actions on $k$-graphs and to higher rank Doplicher-Roberts algebras.

Localization operators related to Stockwell transforms on locally compact abelian groups

Localization operators related to Stockwell transforms on locally compact abelian groups

On logarithms of measures

Let A be a Banach algebra and let xin A have the property that its spectrum does not separate 0 from infinity. It is well known that x has a logarithm, i.e., there exists yin A with x=e^y. We will use this statement to identify measures defined on a locally compact group to have logarithms. Also, we will show that the converse of the above statement is in general not true. Our results will be related to infinitely divisible probability measures.

Sobolev algebras on Lie groups and noncommutative geometry

We show that there exists a quantum compact metric space which underlies the setting of each Sobolev algebra associated to a subelliptic Laplacian \Delta=-(X_1^2+\cdots+X_m^2) on a compact connected Lie group G if p is large enough, more precisely under the (sharp) condition p > \frac{d}{\alpha} , where d is the local dimension of (G,X) and where 0 < \alpha \leq 1 . We also provide locally compact variants of this result and generalizations for real second-order subelliptic operators. We also introduce a compact spectral triple (= noncommutative manifold) canonically associated to each subelliptic Laplacian on a compact group. In addition, we show that its spectral dimension is equal to the local dimension of (G,X) . Finally, we prove that the Connes spectral pseudo-metric allows us to recover the Carnot–Carathéodory distance.

Non-commutative ambits and equivariant compactifications

We prove that an action \rho:A\to M(C_0(\mathbb{G})\otimes A) of a locally compact quantum group on a C^* -algebra has a universal equivariant compactification and prove a number of other category-theoretic results on \mathbb{G} -equivariant compactifications: that the categories compactifications of \rho and A , respectively, are locally presentable (hence complete and cocomplete), that the forgetful functor between them is a colimit-creating left adjoint, and that epimorphisms therein are surjective and injections are regular monomorphisms. When \mathbb{G} is regular, coamenable we also show that the forgetful functor from unital \mathbb{G} - C^* -algebras to unital C^* -algebras creates finite limits and is comonadic and that the monomorphisms in the former category are injective.

Fixed point sets and the fundamental group II: Euler characteristics

For a finite group $G$ of not prime power order, Oliver showed that the obstruction for a finite CW-complex $F$ to be the fixed point set of a contractible finite $G$ -CW-complex is determined by the Euler characteristic $\chi (F)$ . (He also has similar results for compact Lie group actions.) We show that the analogous problem for $F$ to be the fixed point set of a finite $G$ -CW-complex of some given homotopy type is still determined by the Euler characteristic. Using trace maps on $K_0$ [2, 7, 18], we also see that there are interesting roles for the fundamental group and the component structure of the fixed point set.

On Bohr compactifications and profinite completions of group extensions

AbstractLet $G= N\rtimes H$ be a locally compact group which is a semi-direct product of a closed normal subgroup N and a closed subgroup H. The Bohr compactification ${\rm Bohr}(G)$ and the profinite completion ${\rm Prof}(G)$ of G are, respectively, isomorphic to semi-direct products $Q_1 \rtimes {\rm Bohr}(H)$ and $Q_2 \rtimes {\rm Prof}(H)$ for appropriate quotients $Q_1$ of ${\rm Bohr}(N)$ and $Q_2$ of ${\rm Prof}(N).$ We give a precise description of $Q_1$ and $Q_2$ in terms of the action of H on appropriate subsets of the dual space of N. In the case where N is abelian, we have ${\rm Bohr}(G)\cong A \rtimes {\rm Bohr}(H)$ and ${\rm Prof}(G)\cong B \rtimes {\rm Prof}(H),$ where A (respectively B) is the dual group of the group of unitary characters of N with finite H-orbits (respectively with finite image). Necessary and sufficient conditions are deduced for G to be maximally almost periodic or residually finite. We apply the results to the case where $G= \Lambda\wr H$ is a wreath product of discrete groups; we show in particular that, in case H is infinite, ${\rm Bohr}(\Lambda\wr H)$ is isomorphic to ${\rm Bohr}(\Lambda^{\rm Ab}\wr H)$ and ${\rm Prof}(\Lambda\wr H)$ is isomorphic to ${\rm Prof}(\Lambda^{\rm Ab} \wr H),$ where $\Lambda^{\rm Ab}=\Lambda/ [\Lambda, \Lambda]$ is the abelianisation of $\Lambda.$ As examples, we compute ${\rm Bohr}(G)$ and ${\rm Prof}(G)$ when G is a lamplighter group and when G is the Heisenberg group over a unital commutative ring.

Homomorphisms of L1 algebras and Fourier algebras

We investigate conditions for the extendibility of continuous algebra homomorphisms ϕ from the Fourier algebra A(F) of a locally compact group F to the Fourier-Stieltjes algebra B(G) of a locally compact group G to maps between the corresponding L∞ algebras which are weak* continuous. When ϕ is completely bounded and F is amenable, it is induced by a piecewise affine map α:Y→F where Y⊆G. We show that extendibility of ϕ is equivalent to α being an open map. We also study the dual problem for contractive homomorphisms ϕ:L1(F)→M(G). We show that ϕ induces a w* continuous homomorphism between the von Neumann algebras of the groups if and only if the naturally associated map θ (Greenleaf [1965], Stokke [2011]) is a proper map.

Projectivity of some Banach modules over semiprime Banach algebras

Let [Formula: see text] be a semiprime Banach algebra and let [Formula: see text] be a nonzero character on [Formula: see text]. In this paper, we discuss the projectivity of certain left Banach [Formula: see text]-modules in terms of left [Formula: see text]-contractibility of Banach algebras. We also apply our results for the category of quantum group algebras and eventually, we present illustrative examples of the projective left Banach modules over the group algebra [Formula: see text] and the Fourier algebra [Formula: see text] of a locally compact group [Formula: see text].

Littlewood–Paley–Rubio de Francia inequality for multi‐parameter Vilenkin systems

AbstractA version of Littlewood–Paley–Rubio de Francia inequality for bounded multi‐parameter Vilenkin systems is proved: For any family of disjoint sets such that are intervals in and a family of functions with Vilenkin–Fourier spectrum inside the following holds: where C does not depend on the choice of rectangles or functions . This result belongs to the line of studying of (multi‐parameter) generalizations of Rubio de Francia inequality to locally compact abelian groups. The arguments are mainly based on the atomic theory of multi‐parameter martingale Hardy spaces and, as a byproduct, yield an easy‐to‐use multi‐parameter version of Gundy's theorem on the boundedness of operators taking martingales to measurable functions. Additionally, some extensions and corollaries of the main result are obtained, including a weaker version of the inequality for exponents and an example of a one‐parameter inequality for an exotic notion of interval.

Multiplicative Characters and Gaussian Fluctuation Limits

It is known that extreme characters of several inductive limits of compact groups exhibit multiplicativity in a certain sense. In the paper, we formulate such multiplicativity for inductive limit quantum groups and provide explicit examples of multiplicative characters in the case of quantum unitary groups. Furthermore, we show a Gaussian fluctuation limit theorem using this concept of multiplicativity.

Compact groups from semi-analytical models of galaxy formation – IV: Effect of group assembly on the evolution of their galaxies

ABSTRACT Using over 3000 compact groups (CGs) of galaxies extracted from mock catalogues built from semi-analytical models (SAMs) of galaxy formation, we study whether the CG assembly channel affects the z = 0 properties of galaxies and their evolution. The evolution of CG galaxy properties with time is a clear function of their stellar masses. For instance, high-stellar-mass CG galaxies have lived their last 8 Gyr with little cold gas content while maintaining their reservoir of hot gas, while low-mass CG galaxies still preserve some of their cold gas content at the present but they have completely drained their hot gas reservoir. Beyond that, we find that the evolution of CG galaxies is also a function of the assembly history of the CGs: with more extreme losses of gas content, faster mass gain rates for black holes and more marked suppression of star formation as a function of cosmic time as we go from recent to early CG assembly. Thus, CGs constitute another laboratory for galaxy assembly bias, as the later assembling groups have later star formation. Our results show that classifying CGs according to their assembly channel is a way of distinguishing different paths by which galaxies transform their properties throughout their history.

ED-2: A cold but not so narrow stellar stream crossing the solar neighbourhood

Context. ED-2 is a stellar stream identified as a compact group in integrals-of-motion space in a local sample of halo stars from the third Gaia data release. Aims. We investigate its nature and possible association with known halo substructures. Methods. We explored the current properties of ED-2 members in phase-space and also analysed the expected distribution via orbit integration. In addition, we studied the metallicity of ED-2 using APOGEE DR17 and LAMOST DR8 (and re-calibrated DR3). Results. ED-2 forms a compact group in the x − z (or R − z) plane, showing a pancake-like structure as it crosses the solar neighbourhood. Dynamically, it is most similar to the globular clusters NGC 3201 and NGC 6101 and to the stellar streams Ylgr and Phlegethon. However, its orbit is sufficiently different for none of these objects to likely be the ED-2 progenitor. We also find ED-2 to be quite metal poor: all of its stars have [Fe/H]≤ − 2.42, with a median [Fe/H] = −2.60−0.21+0.20. At this low metallicity, it is unlikely that ED-2 stems from any known globular cluster. Instead, ED-2 seems to be in a similar category as the recently discovered Phoenix and C-19 stellar streams. We find that ED-2 members are scattered across the whole sky, which is due to its current orbital phase. We predict that as this object moves to its next apocentre, it will acquire an on-sky morphology that is similar to that of cold stellar streams. Finally, because ED-2 is nearing pericentre, we predict that additional members found below the plane will have high radial velocities of close to ∼500 km s−1 in the present-day direction of the globular cluster NGC 6101.

Numerical investigation of the flow characteristics of supercritical carbon dioxide in a high-speed rotating annular gap

The flow characteristics of Taylor–Couette–Poiseuille flow induced by supercritical carbon dioxide in an annular gap play a pivotal role in determining the overall performance of the rotating machinery. To accurately design the structural components of rotating machinery and enhance its efficiency, this study employs the large eddy simulation method to investigate the flow behavior of Taylor–Couette–Poiseuille flow with supercritical carbon dioxide within an annular gap. The results reveal that vortices are predominantly generated near the inner wall. Initially, the flow exhibits small swirl vortices, spiral ring vortices, and annular vortices along the flow direction. As the flow progresses, these small vortices at the inlet region transition into hairpin swirl vortices. Finally, turbulent flow disturbances lead to the fragmentation and merging of spiral and annular vortices, resulting in a flow field characterized by high-frequency hairpin swirl vortices and small vortices with strong randomness. An increase in the swirl number causes the initial position of the Taylor vortex to shift toward the inlet, while the turbulent kinetic energy is more active on the outer wall side than the inner wall side. Along the flow direction, the vortices experience a developmental process involving stabilization, diffusion, and mixing. Varying the radius ratio affects the magnitude of vorticity, reduces velocity fluctuations in a regular pattern, and alters the distribution of helicity bands from wide and sparse to compact and dense groupings. As the axial Reynolds number increases, the magnitude of vortices grows, leading to more severe velocity fluctuations and the transformation of the helicity bands from a regular annular pattern to fluctuating vortices bands, accompanied by a decrease in helicity.

Harmonic maps and biharmonic maps for double fibrations of compact Lie groups

Harmonic maps and biharmonic maps for double fibrations of compact Lie groups

Strong Pedersen rigidity for coactions of compact groups

We prove a version of Pedersen’s outer conjugacy theorem for coactions of compact groups, which characterizes outer conjugate coactions of a compact group in terms of properties of the dual actions. In fact, we show that every isomorphism of a dual action comes from a unique outer conjugacy of a coaction, which in this context should be called strong Pedersen rigidity. We promote this to a category equivalence.

The Dynamical Systems (A, G, ω) and Pseudo-differential Operators on Locally Compact Commutative Groups

The Dynamical Systems (A, G, ω) and Pseudo-differential Operators on Locally Compact Commutative Groups

Mackey's obstruction map for discrete graded algebras

G.W. Mackey's celebrated obstruction theory for projective representations of locally compact groups was remarkably generalized by J. M. G. Fell and R. S. Doran to the wide area of saturated Banach *-algebraic bundles.Analogous obstruction is suggested here for discrete group graded algebras which are not necessarily saturated, i.e. strongly graded in the discrete context. The discrete obstruction is a map assigning a certain second cohomology class to every equivariance class of absolutely graded-simple modules.The set of equivariance classes of such modules is equipped with an appropriate multiplication, namely a graded product, such that the obstruction map is a homomorphism of abelian monoids. Graded products, essentially arising as pull-backs of bundles, admit many nice properties, including a way to twist graded algebras and their graded modules.The obstruction class turns out to determine the fine part that appears in the Bahturin-Zaicev-Sehgal decomposition, i.e. the graded Artin-Wedderburn theorem for graded-simple algebras which are graded Artinian, in case where the base algebras (i.e. the unit fiber algebras) are finite-dimensional over algebraically closed fields.

Generalized positive energy representations of groups of jets

Let V be a finite-dimensional real vector space and K a compact simple Lie group with Lie algebra \mathfrak{k} . Consider the Fréchet–Lie group G := J_0^\infty(V; K) of \infty -jets at 0\in V of smooth maps V \to K , with Lie algebra \mathfrak{g}=J_0^\infty(V; \mathfrak{k}) . Let P be a Lie group and write \mathfrak{p}:=\operatorname{Lie}(P) . Let \alpha be a smooth P -action on G . We study smooth projective unitary representations \bar{\rho} of G\rtimes_\alpha P that satisfy a so-called generalized positive energy condition. In particular, this class captures representations that are in a suitable sense compatible with a KMS state on the von Neumann algebra generated by \bar{\rho}(G) . We show that this condition imposes severe restrictions on the derived representation d\bar{\rho} of \mathfrak{g}\rtimes \mathfrak{p} , leading in particular to sufficient conditions for {\bar{\rho}}|_{G} to factor through J_0^2(V; K) , or even through K .