LIFTING HAMILTONIAN LOOPS TO ISOTOPIES IN FIBRATIONS

Let G be a Lie group, H a closed subgroup and M the homogeneous space G/H. Each representation Ψ of H determines a G-equivariant principal bundle [Formula: see text] on M endowed with a G-invariant connection. We consider subgroups [Formula: see text] of the diffeomorphism group Diff (M), such that, each vector field [Formula: see text] admits a lift to a preserving connection vector field on [Formula: see text]. We prove that [Formula: see text]. This relation is applicable to subgroups [Formula: see text] of the Hamiltonian groups of the flag varieties of a semisimple group G. Let MΔ be the toric manifold determined by the Delzant polytope Δ. We put φb for the loop in the Hamiltonian group of MΔ defined by the lattice vector b. We give a sufficient condition, in terms of the mass center of Δ, for the loops φb and [Formula: see text] to be homotopically inequivalent.

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Fundamental Vector Fields

Introductory Lectures on Equivariant Cohomology ◽

10.23943/princeton/9780691191751.003.0011 ◽

2020 ◽

pp. 87-96

Author(s):

Loring W. Tu

Keyword(s):

Vector Field ◽

Fixed Points ◽

Lie Algebra ◽

Group Action ◽

Lie Group ◽

Vector Fields ◽

Principal Bundle ◽

Left Action ◽

Fundamental Vector ◽

Vertical Tangent

This chapter addresses fundamental vector fields. The concept of a connection on a principal bundle is essential in the construction of the Cartan model. To define a connection on a principal bundle, one first needs to define the fundamental vector fields. When a Lie group acts smoothly on a manifold, every element of the Lie algebra of the Lie group generates a vector field on the manifold called a fundamental vector field. On a principal bundle, the fundamental vectors are precisely the vertical tangent vectors. In general, there is a relation between zeros of fundamental vector fields and fixed points of the group action. Unless specified otherwise (such as on a principal bundle), a group action is assumed to be a left action.

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SUPERMETRICS ON SUPERMANIFOLDS

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s021988780800276x ◽

2008 ◽

Vol 05 (02) ◽

pp. 271-286 ◽

Cited By ~ 6

Author(s):

G. SARDANASHVILY

Keyword(s):

Gauge Theory ◽

Lie Group ◽

Closed Subgroup ◽

Principal Bundle ◽

Riemannian Metrics ◽

Global Section ◽

General Linear ◽

Base Manifold ◽

General Linear Supergroup ◽

Higgs Fields

By virtue of the well-known theorem, a structure Lie group K of a principal bundle P → X is reducible to its closed subgroup H iff there exists a global section of the quotient bundle P/K → X. In gauge theory, such sections are treated as Higgs fields, exemplified by pseudo-Riemannian metrics on a base manifold X. Under some conditions, this theorem is extended to principal superbundles in the category of G-supermanifolds. Given a G-supermanifold M and a graded frame superbundle over M with a structure general linear supergroup, a reduction of this structure supergroup to an orthogonal-symplectic supersubgroup is associated to a supermetric on a G-supermanifold M.

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THE FERMI–WALKER DERIVATIVE IN LIE GROUPS

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887813200119 ◽

2013 ◽

Vol 10 (07) ◽

pp. 1320011 ◽

Cited By ~ 2

Author(s):

FATMA KARAKUŞ ◽

YUSUF YAYLI

Keyword(s):

Vector Field ◽

Lie Groups ◽

Lie Group ◽

Necessary Conditions ◽

Rotating Frame ◽

Darboux Vector

In this study, Fermi–Walker derivative, Fermi–Walker parallelism, non-rotating frame, Fermi–Walker termed Darboux vector concepts are given for Lie groups in E4. First, we get any Frénet curve and any vector field along the Frénet curve in a Lie group. Then, the Fermi–Walker derivative is defined for the Lie group. Fermi–Walker derivative and Fermi–Walker parallelism are analyzed in Lie groups. Finally, the necessary conditions for Fermi–Walker parallelism are explained.

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Poisson Principal Bundles

Symmetry Integrability and Geometry Methods and Applications ◽

10.3842/sigma.2021.006 ◽

2021 ◽

Author(s):

Shahn Majid ◽

◽

Liam Williams ◽

Keyword(s):

Quantum Group ◽

Lie Group ◽

Poisson Structure ◽

Differential Calculus ◽

Principal Bundle ◽

Poisson Manifold ◽

Spin Connection ◽

Poisson Geometry ◽

Hopf Fibration ◽

Principal Bundles

We semiclassicalise the theory of quantum group principal bundles to the level of Poisson geometry. The total space X is a Poisson manifold with Poisson-compatible contravariant connection, the fibre is a Poisson-Lie group in the sense of Drinfeld with bicovariant Poisson-compatible contravariant connection, and the base has an inherited Poisson structure and Poisson-compatible contravariant connection. The latter are known to be the semiclassical data for a quantum differential calculus. The theory is illustrated by the Poisson level of the q-Hopf fibration on the standard q-sphere. We also construct the Poisson level of the spin connection on a principal bundle.

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CRITERION OF PROPER ACTIONS FOR 3-STEP NILPOTENT LIE GROUPS

International Journal of Mathematics ◽

10.1142/s0129167x07004333 ◽

2007 ◽

Vol 18 (07) ◽

pp. 783-795 ◽

Cited By ~ 5

Author(s):

TARO YOSHINO

Keyword(s):

Homogeneous Space ◽

Lie Groups ◽

Lie Group ◽

Closed Subgroup ◽

Nilpotent Lie Groups ◽

Nilpotent Lie Group ◽

Proper Actions ◽

Affirmative Solution

For a nilpotent Lie group G and its closed subgroup L, Lipsman [13] conjectured that the L-action on some homogeneous space of G is proper in the sense of Palais if and only if the action is free. Nasrin [14] proved this conjecture assuming that G is a 2-step nilpotent Lie group. However, Lipsman's conjecture fails for the 4-step nilpotent case. This paper gives an affirmative solution to Lipsman's conjecture for the 3-step nilpotent case.

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Basic Forms

Introductory Lectures on Equivariant Cohomology ◽

10.23943/princeton/9780691191751.003.0012 ◽

2020 ◽

pp. 97-102

Author(s):

Loring W. Tu

Keyword(s):

Lie Group ◽

Principal Bundle ◽

Differential Forms ◽

Total Space ◽

Lie Derivative ◽

Connected Lie Group

This chapter describes basic forms. On a principal bundle π‎: P → M, the differential forms on P that are pullbacks of forms ω‎ on the base M are called basic forms. The chapter characterizes basic forms in terms of the Lie derivative and interior multiplication. It shows that basic forms on a principal bundle are invariant and horizontal. To understand basic forms better, the chapter considers a simple example. The plane ℝ2 may be viewed as the total space of a principal ℝ-bundle. A connected Lie group is generated by any neighborhood of the identity. This example shows the necessity of the connectedness hypothesis.

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Closure Theorem for Analytic Subgroups of Real Lie Groups

Canadian Mathematical Bulletin ◽

10.4153/cmb-1976-065-9 ◽

1976 ◽

Vol 19 (4) ◽

pp. 435-439 ◽

Cited By ~ 3

Author(s):

D. Ž. Djoković

Keyword(s):

Lie Groups ◽

Lie Group ◽

Closed Subgroup ◽

Main Result Theorem ◽

Closure Theorem ◽

Result Theorem ◽

Analytic Subgroup

Let G be a real Lie group, A a closed subgroup of G and B an analytic subgroup of G. Assume that B normalizes A and that AB is closed in G. Then our main result (Theorem 1) asserts that .This result generalizes Lemma 2 in the paper [4], G. Hochschild has pointed out to me that the proof of that lemma given in [4] is not complete but that it can be easily completed.

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On orbits of unipotent flows on homogeneous spaces, II

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700003382 ◽

1986 ◽

Vol 6 (2) ◽

pp. 167-182 ◽

Cited By ~ 34

Author(s):

S. G. Dani

Keyword(s):

Invariant Subspace ◽

Lebesgue Measure ◽

Compact Subset ◽

Lie Group ◽

Diophantine Approximation ◽

Closed Subgroup ◽

Homogeneous Spaces ◽

Semisimple Lie Group ◽

Unipotent Subgroup ◽

Unipotent Flows

AbstractWe show that if (ut) is a one-parameter subgroup of SL (n, ℝ) consisting of unipotent matrices, then for any ε > 0 there exists a compact subset K of SL(n, ℝ)/SL(n, ℤ) such that the following holds: for any g ∈ SL(n, ℝ) either m({t ∈ [0, T] | utg SL (n, ℤ) ∈ K}) > (1 – ε)T for all large T (m being the Lebesgue measure) or there exists a non-trivial (g−1utg)-invariant subspace defined by rational equations.Similar results are deduced for orbits of unipotent flows on other homogeneous spaces. We also conclude that if G is a connected semisimple Lie group and Γ is a lattice in G then there exists a compact subset D of G such that for any closed connected unipotent subgroup U, which is not contained in any proper closed subgroup of G, we have G = DΓ U. The decomposition is applied to get results on Diophantine approximation.

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Сurvature-torsion tensor for Cartan connection

Differential Geometry of Manifolds of Figures ◽

10.5922/0321-4796-2019-50-18 ◽

2019 ◽

pp. 155-168

Author(s):

Yu. Shevchenko

Keyword(s):

Homogeneous Space ◽

Lie Group ◽

Principal Bundle ◽

Torsion Tensor ◽

Projective Connection ◽

Cartan Connection ◽

Bianchi Identities ◽

Covariant Derivatives ◽

Structure Equations ◽

Derivatives Of

A Lie group containing a subgroup is considered. Such a group is a principal bundle, a typical fiber of this principal bundle is the subgroup and a base is a homogeneous space, which is obtained by factoring the group by the subgroup. Starting from this group, we constructed structure equations of a space with Cartan connection, which generalizes the Cartan point projective connection, Akivis’s linear projective connection, and a plane projective connection. Structure equations of this Cartan connection, containing the components of the curvature-torsion object, allowed: 1) to show that the curvature-torsion object forms a tensor containing a torsion tensor; 2) to find an analogue of the Bianchi identities such that the curvature-torsion tensor and its Pfaff derivatives satisfy this analogue; 3) to obtain the conditions for the transformation of Pfaffian derivatives of the curvature-torsion tensor into covariant derivatives with respect to the Cartan connection.

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Variants of stability of discontinuous groups for Euclidean motion groups

International Journal of Mathematics ◽

10.1142/s0129167x1750046x ◽

2017 ◽

Vol 28 (06) ◽

pp. 1750046 ◽

Cited By ~ 1

Author(s):

Ali Baklouti ◽

Souhail Bejar

Keyword(s):

Lie Group ◽

Closed Subgroup ◽

Deformation Parameter ◽

Motion Group ◽

Discontinuous Group ◽

Euclidean Motion Group ◽

Euclidean Motion ◽

Motion Groups ◽

Discontinuous Groups ◽

Properly Discontinuous Action

Let [Formula: see text] be a Lie group, [Formula: see text] a closed subgroup of [Formula: see text] and [Formula: see text] a discontinuous group for the homogeneous space [Formula: see text]. Given a deformation parameter [Formula: see text], the deformed subgroup [Formula: see text] may fail to act properly discontinuously on [Formula: see text]. To understand this phenomenon in the case when [Formula: see text] stands for an Euclidean motion group [Formula: see text], we compare the notion of stability for discontinuous groups (cf. [T. Kobayashi and S. Nasrin, Deformation of properly discontinuous action of [Formula: see text] on [Formula: see text], Int. J. Math. 17 (2006) 1175–1193]) with its variants. We prove that the defined stability variants hold when [Formula: see text] turns out to be a crystallographic subgroup of [Formula: see text].

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