Showing papers on "Equivariant map published in 1991"
01 Jan 1991
TL;DR: The Structure of the Book as discussed by the authors is a collection of essays about algebraic groups over arbitrary fields, including a discussion of the relation between the structure of closed subgroups and property (T) of normal subgroups.
Abstract: 1. Statement of Main Results.- 2. Synopsis of the Chapters.- 3. Remarks on the Structure of the Book, References and Notation.- 1. Preliminaries.- 0. Notation, Terminology and Some Basic Facts.- 1. Algebraic Groups Over Arbitrary Fields.- 2. Algebraic Groups Over Local Fields.- 3. Arithmetic Groups.- 4. Measure Theory and Ergodic Theory.- 5. Unitary Representations and Amenable Groups.- II. Density and Ergodicity Theorems.- 1. Iterations of Linear Transformations.- 2. Density Theorems for Subgroups with Property (S)I.- 3. The Generalized Mautner Lemma and the Lebesgue Spectrum.- 4. Density Theorems for Subgroups with Property (S)II.- 5. Non-Discrete Closed Subgroups of Finite Covolume.- 6. Density of Projections and the Strong Approximation Theorem.- 7. Ergodicity of Actions on Quotient Spaces.- III. Property (T).- 1. Representations Which Are Isolated from the Trivial One-Dimensional Representation.- 2. Property (T) and Some of Its Consequences. Relationship Between Property (T) for Groups and for Their Subgroups.- 3. Property (T) and Decompositions of Groups into Amalgams.- 4. Property (R).- 5. Semisimple Groups with Property (T).- 6. Relationship Between the Structure of Closed Subgroups and Property (T) of Normal Subgroups.- IV. Factor Groups of Discrete Subgroups.- 1. b-metrics, Vitali's Covering Theorem and the Density Point Theorem.- 2. Invariant Algebras of Measurable Sets.- 3. Amenable Factor Groups of Lattices Lying in Direct Products.- 4. Finiteness of Factor Groups of Discrete Subgroups.- V. Characteristic Maps.- 1. Auxiliary Assertions.- 2. The Multiplicative Ergodic Theorem.- 3. Definition and Fundamental Properties of Characteristic Maps.- 4. Effective Pairs.- 5. Essential Pairs.- VI. Discrete Subgroups and Boundary Theory.- 1. Proximal G-Spaces and Boundaries.- 2. ?-Boundaries.- 3. Projective G-Spaces.- 4. Equivariant Measurable Maps to Algebraic Varieties.- VII. Rigidity.- 1. Auxiliary Assertions.- 2. Cocycles on G-Spaces.- 3. Finite-Dimensional Invariant Subspaces.- 4. Equivariant Measurable Maps and Continuous Extensions of Representations.- 5. Superrigidity (Continuous Extensions of Homomorphisms of Discrete Subgroups to Algebraic Groups Over Local Fields).- 6. Homomorphisms of Discrete Subgroups to Algebraic Groups Over Arbitrary Fields.- 7. Strong Rigidity (Continuous Extensions of Isomorphisms of Discrete Subgroups).- 8. Rigidity of Ergodic Actions of Semisimple Groups.- VIII. Normal Subgroups and "Abstract" Homomorphisms of Semisimple Algebraic Groups Over Global Fields.- 1. Some Properties of Fundamental Domains for S-Arithmetic Subgroups.- 2. Finiteness of Factor Groups of S-Arithmetic Subgroups.- 3. Homomorphisms of S-Arithmetic Subgroups to Algebraic Groups.- IX. Arithmeticity.- 1. Statement of the Arithmeticity Theorems.- 2. Proof of the Arithmeticity Theorems.- 3. Finite Generation of Lattices.- 4. Consequences of the Arithmeticity Theorems I.- 5. Consequences of the Arithmeticity Theorems II.- 6. Arithmeticity, Volume of Quotient Spaces, Finiteness of Factor Groups, and Superrigidity of Lattices in Semisimple Lie Groups.- 7. Applications to the Theory of Symmetric Spaces and Theory of Complex Manifolds.- Appendices.- A. Proof of the Multiplicative Ergodic Theorem.- B. Free Discrete Subgroups of Linear Groups.- C. Examples of Non-Arithmetic Lattices.- Historical and Bibliographical Notes.- References.
1,520 citations
TL;DR: In this article, a model for the differential graded algebra (dga) of differential forms on the free loop space LX of a smooth manifold X and construct certain important differential forms in terms of this model was presented.
144 citations
01 Jan 1991
TL;DR: In this article, an action σ: G × P→P of a Poisson Lie group G on a smooth manifold P is defined as an action that preserves the Poisson structure on P. If P is symplectic and if σ is generated by an equivariant momentum mapping J: P→ g*, the reduction procedure of Meyer [Me] and Marsden and Weinstein [Ms-We] gives a way of describing the symplectic leaves of G \ P as the quotients P µ := G µ \J −1 (µ), where µ�
Abstract: An action σ: G × P→P of a Poisson Lie group G on a Poisson manifold P is called a Poisson action if σ is a Poisson map. It is believed that Poisson actions should be used to understand the “hidden symmetries” of certain integrable systems [STS2]. If the Poisson Lie group G has the zero Poisson structure, then σ being a Poisson action is equivalent to each transformation σ g : P→ P for g ∈ G preserving the Poisson structure on P. In this case, if the orbit space G \ P is a smooth manifold, it has a reduced Poisson structure such that the projection map P→G \ P is a Poisson map. If P is symplectic and if the action σ is generated by an equivariant momentum mapping J: P→ g*, the reduction procedure of Meyer [Me] and Marsden and Weinstein [Ms-We] gives a way of describing the symplectic leaves of G \ P as the quotients P µ := G µ \J −1 (µ), where µ∈ g* and G µ ⊂ G is the coadjoint isotropy subgroup of µ.
124 citations
TL;DR: In this paper, the fixed point set F(EG, X) G denotes the function space of all maps EG ~ X, equipped with a G-action by 9f = 9f9-1.
Abstract: Let G be a p-group, and let X be a G-complex Let EG denote a contractible space on which G acts freely By the "homotopy fixed point set" of X, we mean the fixed point set F(EG, X) G, where F(EG, X) denotes the function space of all maps EG ~ X, equipped with a G-action by 9f = 9f9-1 If we let * denote the one point space with trivial G-action, we may also consider the G-space F ( , ,X ) ; it is canonically G-homeomorphic to X The G-map E G ~ , induces a map ~/: X G ~F( , , X) G In [19], D Sullivan proposed the following
46 citations
TL;DR: The median is equivariant, monotonic, and has 50% breakdown as mentioned in this paper, which is the first estimator that has all three properties and no other estimator has them.
Abstract: The median is equivariant, monotonic, and has 50% breakdown. No other estimator has all three properties.
41 citations
TL;DR: In this paper, it was shown that the equivariant version of BF is equivalent to the system of fixed sets of a G-space, i.e., it behaves for-mally like the fixed set system of a fixed-set system of an infinite loop space.
Abstract: We develop machinery enabling us to show that suitable G-spaces, including the equivariant version of BF , are equivariant infinite loop spaces. This involves a "recognition principle" for systems of spaces which behave for- mally like the system of fixed sets of a G-space; that is, we give a necessary and sufficient condition for such a system to be equivalent to the fixed set system of an equivariant infinite loop space. The advantage of using the language of fixed set systems is that one can frequently replace the system of fixed sets of an ac- tual G-space by an equivalent formal system which is considerably simpler, and which admits the requisite geometry necessary for delooping. We also apply this machinery to construct equivariant Eilenberg-Mac Lane spaces corresponding to Mackey functors.
41 citations
TL;DR: In this paper, the authors show that the Hermitian type and Riemannian symmetric spaces are equivalent and that there exists an open regular H-invariant cone in q =h such that k ∩ Ω ≠ 0.
Abstract: Let M =G/H be a semisimple symmetric space,τ the corresponding involution and D =G/K the Riemannian symmetric space. Then we show that the followingare equivalent: M is of Hermitian type; τ induces a conjugation on D; thereexists an open regular H-invariant cone Ω in q =h[bottom] such that k ∩ Ω ≠ 0. We relate the spaces of Hermitian type to the regular and parahermitian symmetric spaces, analyze the fine structure of D under τ and construct an equivariant Cayley transform. We collect also some results on the classification of invariant cones in q. Finally we point out some applications in representations theory.
39 citations
TL;DR: In this article, it was shown that all the known results on the preservation of first integrals by a symplectic difference scheme can be unified under the equivalence theorem of Feng-Ge.
37 citations
28 citations
01 Jul 1991
TL;DR: In this article, it was shown that relative equilibria and relative periodic orbits of G-equivariant vector fields correspond to equilibra and periodic orbits, respectively, of vector fields.
Abstract: Dynamics in a neighbourhood of a hyperbolic equilibrium or periodic orbit are wellunderstood. Thus, if XQ is a hyperbolic equilibrium point, there exist smooth stable and unstable manifolds through XQ and the local flowof X near XQ is topologically conjugate to that of the linearized flow x = DX(xo)(x) (Hartman's theorem). If / is a periodic orbit, the hyperbolicity of / may be described in terms of either the Poincare map or the Floquet exponents. If / is hyperbolic, we again have stable and unstable manifolds through / and a version of Hartman's theorem conjugating the local flow to the flow linearized in the normal bundle of /. Suppose now that G is a compact Lie group acting smoothly on M and that X is a G-equivariant vector field on M with associated G-equivariant flow q;x. We recall that a group orbit a C M is called a relative equilibrium of X if a is a q;X-invariant subset of M. If E C M is a compact q;X-invariant· subset of M such that (1) EjG S!! Sl ; and (2) q;X induces a non-trivial flow on s-, we call E a relative periodic orbit of X. Relative equilibria and relative periodic orbits of G-equivariant vector fields respectively correspond to equilibria and periodic orbits of (non-equivariant) vector fields. It is easy to show that if / is a maximal trajectory of q;X such that the G-orbit of /, G '/, is compact, then G . / is either a relative equilibrium or a relative periodic orbit of X. Our aim in this work is to obtain generalisations of (1), (2) for relative equilibria and periodic orbits and to describe the corresponding generic local theory. We shall do this for both equivariant diffeomorphisms and vector fields.
27 citations
TL;DR: A complete classification of equivariant surgery on incompressible tori with respect to involutions with possible 1or 2-dimensional fixed sets is given in this article, where a complete classification is given for equivariance on equivariant tori.
Abstract: A complete classification is given for equivariant surgery on incompressible tori with respect to involutions with possible 1or 2-dimensional fixed sets.
TL;DR: In this article, Petrie et al. show that there exists a Baire set in the G-invariant metrics on M such that the moduli space of G of irreducible self-dual connections is a manifold.
Abstract: Let M be a smooth simply connected closed 4-manifold with positive definite intersection form. Suppose a finite group G acts smoothly on M. Let 7r: E -+ M be the instanton number one quaternion line bundle over M with a smooth G-action such that 7r is an equivariant map. We first show that there exists a Baire set in the G-invariant metrics on M such that the moduli space . G of G-invariant irreducible self-dual connections is a manifold. By utilizing the G-transversality theory of T. Petrie, we then identify cohomology obstructions to globally perturbing the full space ./ of irreducible self-dual connections to a G-manifold when G = Z2 and the fixed point set of the Z2 action on M is a nonempty collection of isolated points and Riemann surfaces.
TL;DR: This paper constructs continuous families of nonisomorphic algebraic G-vector bundles in which the base space is a fixed representation of G and shows that in some cases these vector bundles yield continuous Families of distinct G-actions on affine spaces.
Abstract: Let G be a connected semisimple Lie group over C. In this paper we construct continuous families of nonisomorphic algebraic G-vector bundles in which the base space is a fixed representation of G. The G-vector bundles constructed are all G-invariant hypersurfaces in a representation of G. We show that in some cases these vector bundles yield continuous families of distinct G-actions on affine spaces.
TL;DR: In this article, the authors extend the framework of cyclic cohomology to the equivariant context and propose a framework for cyclic cyclic co-occurrences.
Abstract: We extend the framework of entire cyclic cohomology to the equivariant context.
TL;DR: In this paper, the Chern character in the equivariant entire cyclic cohomology was constructed and a general index theorem for the G-invariant Dirae operator was proved.
Abstract: We construct the Chern character in the equivariant entire cyclic cohomology. We prove a general index theorem for the G-invariant Dirae operator.
TL;DR: These constructions and invariants are used to produce families of inequivalent G-vector bundles over G-modules and families of equivariant G actions on affine spaces for some finite and some connected semisimple groups.
Abstract: Let G be a reductive algebraic group and let B be an affine variety with an algebraic action of G. Everything is defined over the field C of complex numbers. Consider the trivial G-vector bundle B x S = S over B where S is a G-module. From the endomorphism ring R of the G-vector bundle S a construction of G-vector bundles over B is given. The bundles constructed this way have the property that when added to S they are isomorphic to F + S for a fixed G-module F. For such a bundle E an invariant rho(E) is defined that lies in a quotient of R. This invariant allows us to distinguish nonisomorphic G-vector bundles. This is applied to the case where B is a G-module and, in that case, an invariant of the underlying equivariant variety is given too. These constructions and invariants are used to produce families of inequivalent G-vector bundles over G-modules and families of inequivalent G actions on affine spaces for some finite and some connected semisimple groups.
TL;DR: In this article, the notions of linearly independent and orthogonal immersions and adjoint hyperquadrics are defined and the relation between linearly indedendent and non-independents is investigated.
Abstract: In this article we define the notions of linearly independent and orthogonal immersions and introduce the notion of adjoint hyperquadrics of linearly independent immersions. We investigate the relations between linearly indedendent immersions, orthogonal immersions, equivariant immersions and adjoint hyperquadrics. Several results in this respect are obtained.
01 Jul 1991
TL;DR: In this article, the authors classified two-parameter bifurcations up to codimension three, using a twoparameter version of parametrised contact equivalence.
Abstract: This thesis contains the classification of two-parameter bifurcations up to codimension three, using a two-parameter version of parametrised contact equivalence.
Part one contains the classification up to codimension one. The result consists of the following components:
1. A list of normal forms for the germs having codimension less or equal to one.
2. Recognition conditions for each normal form in the list, i. e. conditions that characterise the equivalence class of the normal form. These conditions are equations and inequalities for the Taylor coefficients of the germs.
3. Universal unfoldings for each normal form.
The result is obtained by investigating the structure of the orbits, which are induced by the action of the group of equivalences on the space of all bifurcation problems. Techniques from algebra, algebraic geometry and singularity theory are applied.
In part two the classification is extended to codimension three. The second chapter of part two contains a generalisation of the singularity approach to equivariant bifurcation theory. The case of an action of a compact Lie group on state and parameter space is considered. The main example is the case of bifurcations with a certain D4-symmetry.
TL;DR: In this paper, the conjugates of equίvariant holomorphic maps of symmetric domains associated to morphisms of arithmetic varieties are constructed, and it is shown that the corresponding conjugate of a Kuga fiber variety is another Kuga fibre variety.
Abstract: In this paper we construct the conjugates of equίvariant holomorphic maps of symmetric domains associated to morphisms of arithmetic varieties We also prove that the conjugate of a Kuga fiber variety is another Kuga fiber variety
TL;DR: The Lefschetz Theorem in Higher Equivariant K-theory has been studied in this article, where it is shown that higher equivariant ktheory is a higher-order Ktheory.
Abstract: (1991). The Lefschetz Theorem in Higher Equivariant K-theory. Communications in Algebra: Vol. 19, No. 12, pp. 3411-3422.
TL;DR: In this paper, the authors analyse the bifurcations of a general ordinary dififerential equation where is equivariant under an action of the group O(2) on.
Abstract: We analyse the bifurcations of a general ordinary dififerential equation where is equivariant under an action of the group O(2) on. The equation represents the most general nonlinear local interaction of three O(2)-symmetric modes:a steady-state mode with mode-number k, and two periodic (Hopf) modes with mode-numbers l and m. The parameter λ is a bifurcation parameter, and α1, α2are unfolding parameters that split the individual modes apart. The system is assumed to be in Birkhoff normal form, so that f also commutes with an action of the 2-torus T2. We discuss the existence and stability of bifurcating branches and how these break the O(2) × T2 symmetry.Depending on the precise mode-numbers k l m we find up to 31 symmetry classes of possible solutions including six that combine all three modes, and thus cannot be found in any 2-mode interaction. We also discuss the possible occurrence of Sacher-Naimark torus bifurcations, providing a further 10 solution types, and 'slow drift'bifurcations. This 10-dimens...
TL;DR: In this paper, an invariance approach to deriving superior estimators in the Pitman closeness criterion for point estimation in models with group structures is discussed, where the maximal invariant statistic is parameter-free and the closest equivariant estimator to the true value is presented.
Abstract: For the point estimation in models with group structures, an invariance approach to deriving superior estimators is discussed in the Pitman closeness (PC) criterion. When the maximal invariant statistic is parameter-free, that is, ancillary, the closest equivariant estimator to the true value in the PC criterion is presented. On the other hand, as an example where a distribution of the maximalinvariant statistic depends on unknown parameters, the paper treats the Stein problem in estimation of a variance and obtains an improved estimator in the PC criterion by Stein's invariance approach. Also the Stein problem in simultaneous estimation of a location vector of a spherical symmetric distribution is studied.
01 Jul 1991
TL;DR: For a bifurcation germ F(x,λ):ℝn+1,0,0→n,0 which is equivariant with respect to the action of a finite group G, there are permutation actions of G on various subsets of branches of F−1(0) as mentioned in this paper.
Abstract: For a bifurcation germ F(x,λ):ℝn+1,0→ℝn,0 which is equivariant with respect to the action of a finite group G, there are permutation actions of G on various subsets of branches of F−1(0). These sets include the set of all branches as well as the set of branches where λ>0 or 0 or <0, We shall give formulas for the modular characters of these permutation representations (which are the regular characters restricted to the odd order elements of G). These formulas are in terms of the representations of G on certain finite dimensional algebras associated to F. We deduce sufficient conditions for the existence of submaximal orbits by comparing the permutation representations for maximal orbits with certain representations of G.
01 Aug 1991
TL;DR: In this article, it was shown that for any continuous mapf: G n,k→Gn,l the induced map in Z/2-cohomology is either zero in positive dimensions or has image in the subring generated by w1(γn, k), provided 1≤l
Abstract: LetG n,k denote the Grassmann manifold ofk-planes in ℝn. We show that for any continuous mapf: G
n,k→Gn,l the induced map inZ/2-cohomology is either zero in positive dimensions or has image in the subring generated by w1(γn, k), provided 1≤l
TL;DR: In this article, Dold's fixed point index and fixed point transfer are generalized for certain coincidence situations, namely maps which change the "equivariant di-mension." Those invariants change the dimension correspondingly.
Abstract: Dold's fixed point index and fixed point transfer are generalized for certain coincidence situations, namely maps which change the "equivariant di- mension." Those invariants change the dimension correspondingly. A formula for the index of a situation over a space with trivial group action is exhibited. For the transfer, a generalization of Dold's Lefschetz-Hopf trace formula is proved.
01 Jan 1991
TL;DR: In this article, the rational equivariant homotopy type of a topological space X equipped with an action of the group of integers modulo n was studied and the homotopical fixed-point set appeared in the construction of a model of the fixed points set.
Abstract: We study the rational equivariant homotopy type of a topological space X equipped with an action of the group of integers modulo n . For n = pk (p prime, k a positive integer), we build an algebraic model which gives the rational equivariant homotopy type of X. The homotopical fixed-point set appears in the construction of a model of the fixed-points set. In general, this model is different from G. Triantafillou's model [Ti]. For n = p (p prime), we then give a notion of equivariant formality. We prove that this notion is equivalent to the formalizability of the inclusion of fixed-points set i: XzP -X. Examples and counterexamples of Zp-formal spaces are given.