Automorphisms of compact groups

AbstractWe study finitely generated, abelian groups Γ of continuous automorphisms of a compact, metrizable group X and introduce the descending chain condition for such pairs (X, Γ). If Γ acts expansively on X then (X, Γ) satisfies the descending chain condition, and (X, Γ) satisfies the descending chain condition if and only if it is algebraically and topologically isomorphic to a closed, shift-invariant subgroup of GΓ, where G is a compact Lie group. Furthermore every such subgroup of GΓ is a (higher dimensional) Markov shift whose alphabet is a compact Lie group. By using the descending chain condition we prove, for example, that the set of Γ-periodic points is dense in X whenever Γ acts expansively on X. Furthermore, if X is a compact group and (X, Γ) satisfies the descending chain condition, then every ergodic element of Γ has a dense set of periodic points. Finally we give an algebraic description of pairs (X, Γ) satisfying the descending chain condition under the assumption that X is abelian.

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On the existence of ergodic automorphisms in ergodic ℤd-actions on compact groups

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385709000728 ◽

2009 ◽

Vol 30 (6) ◽

pp. 1803-1816 ◽

Cited By ~ 3

Author(s):

C. R. E. RAJA

Keyword(s):

London Math ◽

Abelian Groups ◽

Chain Condition ◽

Compact Groups ◽

Finitely Generated ◽

Finitely Generated Group ◽

Metrizable Group ◽

Compact Abelian Groups ◽

Descending Chain Condition

AbstractLet K be a compact metrizable group and Γ be a finitely generated group of commuting automorphisms of K. We show that ergodicity of Γ implies Γ contains ergodic automorphisms if center of the action, Z(Γ)={α∈Aut(K)∣α commutes with elements of Γ} has descending chain condition. To explain that the condition on the center of the action is not restrictive, we discuss certain abelian groups which, in particular, provide new proofs to the theorems of Berend [Ergodic semigroups of epimorphisms. Trans. Amer. Math. Soc.289(1) (1985), 393–407] and Schmidt [Automorphisms of compact abelian groups and affine varieties. Proc. London Math. Soc. (3) 61 (1990), 480–496].

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Lifting homotopies through fixed points II

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500025336 ◽

1984 ◽

Vol 96 (3-4) ◽

pp. 201-205 ◽

Cited By ~ 1

Author(s):

M. A. Armstrong

Keyword(s):

Normal Subgroup ◽

Fixed Points ◽

Lie Group ◽

Hausdorff Space ◽

Final Section ◽

Invariant Subgroup ◽

Compact Lie Group ◽

Discontinuous Group ◽

Group Of Homeomorphisms ◽

Path Connected

SynopsisThis note complements an earlier paper of the same title. Let G be a discontinuous group of homeomorphisms of a connected, locally path connected, Hausdorff space X, and let ∏:X → X/G denote the associated projection. We work relative to a G-invariant subgroup H of the fundamental group of X and investigate the quotient group ∏1(X/G)/∏*(H). By choosing H appropriately, we can calculate ∏1(X/G) and show that ∏1(X/G)/∏*(∏1(X)) is isomorphic to G/F, where F is the normal subgroup of G generated by those elements which have fixed points. In a final section, we give analogous results for actions of a compact Lie group.

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Rings with descending chain condition on certain principal ideals

Indagationes Mathematicae (Proceedings) ◽

10.1016/1385-7258(77)90070-1 ◽

1977 ◽

Vol 80 (3) ◽

pp. 225-229 ◽

Cited By ~ 4

Author(s):

Kurt Meyberg ◽

Birge Zimmermann-Huisgen

Keyword(s):

Chain Condition ◽

Principal Ideals ◽

Descending Chain Condition

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Construction and Characterization of Representations of SU(7) for GUT Model Builders

Symmetry ◽

10.3390/sym13061044 ◽

2021 ◽

Vol 13 (6) ◽

pp. 1044

Author(s):

Daniel Jones ◽

Jeffery A. Secrest

Keyword(s):

Gauge Symmetry ◽

Symmetry Group ◽

Lie Group ◽

Cartan Subalgebra ◽

Natural Extension ◽

Grand Unification ◽

Raising And Lowering Operators ◽

Higher Dimensional ◽

Lowering Operators

The natural extension to the SU(5) Georgi-Glashow grand unification model is to enlarge the gauge symmetry group. In this work, the SU(7) symmetry group is examined. The Cartan subalgebra is determined along with their commutation relations. The associated roots and weights of the SU(7) algebra are derived and discussed. The raising and lowering operators are explicitly constructed and presented. Higher dimensional representations are developed by graphical as well as tensorial methods. Applications of the SU(7) Lie group to supersymmetric grand unification as well as applications are discussed.

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The Energy Flow on the Loop Space of a Compact Lie Group

Journal of the London Mathematical Society ◽

10.1112/jlms/s2-26.3.557 ◽

1982 ◽

Vol s2-26 (3) ◽

pp. 557-566 ◽

Cited By ~ 18

Author(s):

A. N. Pressley

Keyword(s):

Lie Group ◽

Energy Flow ◽

Loop Space ◽

Compact Lie Group

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RATIONAL LOCAL SYSTEMS AND CONNECTED FINITE LOOP SPACES

Glasgow Mathematical Journal ◽

10.1017/s0017089520000658 ◽

2021 ◽

pp. 1-29

Author(s):

DREW HEARD

Keyword(s):

Lie Group ◽

Loop Space ◽

Algebraic Model ◽

Compact Lie Group ◽

Loop Spaces ◽

Local Systems ◽

Special Case ◽

Finite Loop ◽

Finite Loop Space

Abstract Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group always has an algebraic model. Based on this idea, we show that the category of rational local systems on a connected finite loop space always has a simple algebraic model. When the loop space arises from a connected compact Lie group, this recovers a special case of a result of Pol and Williamson about rational cofree G-spectra. More generally, we show that if K is a closed subgroup of a compact Lie group G such that the Weyl group W G K is connected, then a certain category of rational G-spectra “at K” has an algebraic model. For example, when K is the trivial group, this is just the category of rational cofree G-spectra, and this recovers the aforementioned result. Throughout, we pay careful attention to the role of torsion and complete categories.

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Stratifications of equivariant varieties

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700023315 ◽

1977 ◽

Vol 16 (2) ◽

pp. 279-295 ◽

Cited By ~ 13

Author(s):

M.J. Field

Keyword(s):

Lie Group ◽

General Position ◽

Higher Order ◽

Order Conditions ◽

Compact Lie Group ◽

Algebraic Varieties ◽

Equivariant Maps

Let G be a compact Lie group and V and W be linear G spaces. A study is made of the canonical stratification of some algebraic varieties that arise naturally in the theory of C∞ equivariant maps from V to W. The main corollary of our results is the equivalence of Bierstone's concept of “equivariant general position” with our own of “G transversal”. The paper concludes with a description of Bierstone's higher order conditions for equivariant maps in the framework of equisingularity sequences.

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