Compact Abelian Lie Group Research Articles

Frobenius Distributions of Low Dimensional Abelian Varieties Over Finite Fields

Abstract Given a $g$-dimensional abelian variety $A$ over a finite field $\mathbf{F}_{q}$, the Weil conjectures imply that the normalized Frobenius eigenvalues generate a multiplicative group of rank at most $g$. The Pontryagin dual of this group is a compact abelian Lie group that controls the distribution of high powers of the Frobenius endomorphism. This group, which we call the Serre–Frobenius group, encodes the possible multiplicative relations between the Frobenius eigenvalues. In this article, we classify all possible Serre–Frobenius groups that occur for $g \le 3$. We also give a partial classification for simple ordinary abelian varieties of prime dimension $g\geq 3$.

Modular orbits on the representation spaces of compact abelian Lie groups

Let S be a closed surface of genus g greater than zero. In the present paper, we study the topological-dynamical action of the mapping class group on the \mathbb T^n -character variety giving necessary and sufficient conditions for \mathrm{Mod}(S) -orbits to be dense. As an application, such a characterisation provides a dynamical proof of the Kronecker's theorem concerning inhomogeneous Diophantine approximation.

On the Structure Theory of Cubespace Fibrations

We study fibrations in the category of cubespaces/nilspaces. We show that a fibration of finite degree $$f :X\rightarrow Y$$ between compact ergodic gluing cubespaces (in particular nilspaces) factors as a (possibly countable) tower of compact abelian Lie group principal fiber bundles over Y. If the structure groups of f are connected then the fibers are (uniformly) isomorphic (in a strong sense) to an inverse limit of nilmanifolds. In addition we give conditions under which the fibers of f are isomorphic as subcubespaces. We introduce regionally proximal equivalence relations relative to factor maps between minimal topological dynamical systems for an arbitrary acting group. We prove that any factor map between minimal distal systems is a fibration and conclude that if such a map is of finite degree then it factors as a (possibly countable) tower of principal abelian Lie compact group extensions, thus achieving a refinement of both the Furstenberg’s and the Bronstein–Ellis structure theorems in this setting.

Product Property of Equivariant Degree Under the Action of a Compact Abelian Lie Group

We study local equivariant maps on real finite dimensional orthogonal representations of a compact abelian Lie group G. Equivariant degree $$\deg _G$$ is an invariant applied to determine whether a given map has zeros. The goal of this paper is to present a complete, straightforward proof of the product property of $$\deg _G$$ . For that purpose, we use the otopy classification and distinguish a special kind of map in each class.

On equicontinuous factors of flows on locally path-connected compact spaces

AbstractWe consider a locally path-connected compact metric space K with finite first Betti number $\textrm {b}_1(K)$ and a flow $(K, G)$ on K such that G is abelian and all G-invariant functions $f\,{\in}\, \textrm{C}(K)$ are constant. We prove that every equicontinuous factor of the flow $(K, G)$ is isomorphic to a flow on a compact abelian Lie group of dimension less than ${\textrm {b}_1(K)}/{\textrm {b}_0(K)}$ . For this purpose, we use and provide a new proof for Theorem 2.12 of Hauser and Jäger [Monotonicity of maximal equicontinuous factors and an application to toral flows. Proc. Amer. Math. Soc.147 (2019), 4539–4554], which states that for a flow on a locally connected compact space the quotient map onto the maximal equicontinuous factor is monotone, i.e., has connected fibers. Our alternative proof is a simple consequence of a new characterization of the monotonicity of a quotient map $p\colon K\to L$ between locally connected compact spaces K and L that we obtain by characterizing the local connectedness of K in terms of the Banach lattice $\textrm {C}(K)$ .

FACTORISATION OF EQUIVARIANT SPECTRAL TRIPLES IN UNBOUNDED -THEORY

We provide sufficient conditions to factorise an equivariant spectral triple as a Kasparov product of unbounded classes constructed from the group action on the algebra and from the fixed point spectral triple. We show that if factorisation occurs, then the equivariant index of the spectral triple vanishes. Our results are for the action of compact abelian Lie groups, and we demonstrate them with examples from manifolds and$\unicode[STIX]{x1D703}$-deformations. In particular, we show that equivariant Dirac-type spectral triples on the total space of a torus principal bundle always factorise. Combining this with our index result yields a special case of the Atiyah–Hirzebruch theorem. We also present an example that shows what goes wrong in the absence of our sufficient conditions (and how we get around it for this example).

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.

On the exponent of -spaces and isovariant extensors

The equivariant version of the Curtis-Schori-West theorem is investigated. It is proved that for a nondegenerate Peano -continuum with an action of the compact abelian Lie group , the exponent is equimorphic to the maximal equivariant Hilbert cube if and only if the free part is dense in . We also show that the latter is sufficient for the equimorphy of and in the case of an action of an arbitrary compact Lie group . The key to the proof of these results lies in the theory of the universal -space (in the sense of Palais). Bibliography: 28 titles.

A reconstruction theorem for Connes–Landi deformations of commutative spectral triples

We formulate and prove an extension of Connes’s reconstruction theorem for commutative spectral triples to so-called Connes–Landi or isospectral deformations of commutative spectral triples along the action of a compact Abelian Lie group G , also known as toric noncommutative manifolds. In particular, we propose an abstract definition for such spectral triples, where noncommutativity is entirely governed by a deformation parameter sitting in the second group cohomology of the Pontryagin dual of G , and then show that such spectral triples are well-behaved under further Connes–Landi deformation, thereby allowing for both quantisation from and dequantisation to G -equivariant abstract commutative spectral triples. We then use a refinement of the Connes–Dubois-Violette splitting homomorphism to conclude that suitable Connes–Landi deformations of commutative spectral triples by a rational deformation parameter are almost-commutative in the general, topologically non-trivial sense.

The full orbifold K-theory of abelian symplectic quotients

AbstractIn their 2007 paper, Jarvis, Kaufmann, and Kimura defined the full orbifoldK-theory of an orbifold , analogous to the Chen-Ruan orbifold cohomology of in that it uses the obstruction bundle as a quantum correction to the multiplicative structure. We give an explicit algorithm for the computation of this orbifold invariant in the case when arises as an abelian symplectic quotient. To this end, we introduce the inertial K-theory associated to a T -action on a stably complex manifold M, where T is a compact abelian Lie group. Our methods are integral K-theoretic analogues of those used in the orbifold cohomology case by Goldin, Holm, and Knutson in 2005. We rely on the K-theoretic Kirwan surjectivity methods developed by Harada and Landweber. As a worked class of examples, we compute the full orbifold K-theory of weighted projective spaces that occur as a symplectic quotient of a complex affine space by a circle. Our computations hold over the integers, and in the particular case of these weighted projective spaces, we show that the associated invariant is torsion-free.

Stringy product on twisted orbifold K-theory for abelian quotients

In this paper we present a model to calculate the stringy product on twisted orbifold K-theory of Adem-Ruan-Zhang for abelian complex orbifolds. In the first part we consider the non-twisted case on an orbifold presented as the quotient of a manifold acted by a compact abelian Lie group. We give an explicit description of the obstruction bundle, we explain the relation with the product defined by Jarvis-Kaufmann-Kimura and, via a Chern character map, with the Chen-Ruan cohomology, we explicitly calculate the stringy product for a weighted projective orbifold. In the second part we consider orbifolds presented as the quotient of a manifold acted by a finite abelian group and twistings coming from the group cohomology. We show a decomposition formula for twisted orbifold K-theory that is suited to calculate the stringy product and we use this formula to calculate two examples when the group is (ℤ/2) 3 .

TANGENTIAL REPRESENTATIONS AT ISOLATED FIXED POINTS OF ODD-DIMENSIONAL G-MANIFOLDS

Let G be a compact abelian Lie group, and M an odd-dimensional closed smooth G-manifold. If the fixed point set <TEX>$M^G\neq\emptyset$</TEX> and dim <TEX>$M^G=0$</TEX>, then G has a subgroup H with <TEX>$G/H{\cong}\mathbb{Z}_2$</TEX>, the cyclic group of order 2. The tangential representation <TEX>$\tau_x$</TEX>(M) of G at <TEX>$x{\in}M^G$</TEX> is also regarded as a representation of H by restricted action. We show that the number of fixed points is even, and that the tangential representations at fixed points are pairwise isomorphic as representations of H.

LIMIT FUNCTIONS FOR CONVERGENCE GROUPS AND UNIFORMLY QUASIREGULAR MAPS

We investigate the limit functions of iterates of a function belonging to a convergence group or of a uniformly quasiregular mapping. We show that it is not possible for a subsequence of iterates to tend to a non-constant limit function, and for another subsequence of iterates to tend to a constant limit function. It follows that the closure of the stabiliser of a Siegel domain for a uniformly quasiregular mapping is a compact abelian Lie group, which we further conjecture to be infinite. This result concerning possible limits of convergent subsequences of iterates for holomorphic rational functions on the Riemann sphere is known, and the only known method of proof involves universal covering surfaces and Möbius groups. Hence, our method yields a new and perhaps more elementary proof also in that case.

Nonassociative Tori and Applications to T-Duality

In this paper, we initiate the study of C *-algebras endowed with a twisted action of a locally compact abelian Lie group , and we construct a twisted crossed product , which is in general a nonassociative, noncommutative, algebra. The duality properties of this twisted crossed product algebra are studied in detail, and are applied to T-duality in Type II string theory to obtain the T-dual of a general principal torus bundle with general H-flux, which we will argue to be a bundle of noncommutative, nonassociative tori. Nonassociativity is interpreted in the context of monoidal categories of modules. We also show that this construction of the T-dual includes the other special cases already analysed in a series of papers.

Extending algebraic actions

There is a well-known procedure -induction- for extending an action of a subgroup H of a Lie group G on a topological space X to an action of G on an associated space. Induction can also extend a smooth action of a subgroup H of a Lie group G on a manifold M to a smooth action of G on an associated manifold. In this paper elementary methods are used to show that induction also works in the category of (nonsingular) real algebraic varieties and regular or entire maps if G is a compact abelian Lie group.

Toeplitz operators and weighted Wiener-Hopf operators, pseudoconvex Reinhardt and tube domains

The notion of weighted Wiener-Hopf operators is introduced. Their relationship with Toeplitz operators acting on the space of holomorphic functions which are square integrable with respect to a given "symmetric" measure is discussed. The groupoid approach is used in order to present a general program for studying the C ∗ {C^{\ast } } -algebra generated by weighted Wiener-Hopf operators associated with a solid cone of a second countable locally compact Hausdorff group. This is applied to the case when the group is the dual of a connected locally compact abelian Lie group and the measure is "well behaved" in order to produce a geometric groupoid which is independent of the representation. The notion of a Reinhardt-tube domain Ω \Omega appears thus naturally, and a decomposition series of the corresponding C ∗ {C^{\ast } } -algebra is presented in terms of groupoid C ∗ {C^{\ast } } -algebras associated with various parts of the boundary of the domain Ω \Omega .

Finite homological dimension of 𝐵𝑃_{∗}(𝑋) for infinite complexes

The main result proved here is that B P ∗ ( E G × G X ) {\text {B}}{{\text {P}}_* }(EG{ \times _G}X) has finite homological dimension when G = Z p G = {{\mathbf {Z}}_p} and X X is a finite G - CW G{\text { - CW}} -complex. The argument uses B P ∗ BP {\text {B}}{{\text {P}}_*}{\text {BP}} -comodules.

Differentiable Actions of Compact Abelian Lie Groups on S n

Differentiable Actions of Compact Abelian Lie Groups on S n

Differentiable actions of compact abelian Lie Groups on 𝑆ⁿ

Differentiable actions of compact abelian Lie Groups on 𝑆ⁿ

Equivariant imbeddings of compact abelian Lie groups of transformations

Equivariant imbeddings of compact abelian Lie groups of transformations