On Complex Homogeneous Spaces with Top Homology in Codimension Two

1994 ◽  
Vol 46 (5) ◽  
pp. 897-919 ◽  
Author(s):  
D. N. Akhiezer ◽  
B. Gilligan
Keyword(s):  
Lie Group ◽  
Homology Group ◽  
Principal Bundle ◽  
Homogeneous Spaces ◽  
Codimension Two ◽  
Affine Cone

AbstractDefine dx to be the codimension of the top nonvanishing homology group of the manifold X with coefficients in 2. We investigate homogeneous spaces X := G/H, where G is a connected complex Lie group and H is a closed complex subgroup for which dx = 1,2 and O(X) ≠ ℂ. There exists a fibration π: G/H → G/U such that G/U is holomorphically separable and π*(O(G/U)) = O(G/H), see [11]. We prove the following. If dx = 1, then F := U/H is compact and connected and Y :=G/U is an affine cone with its vertex removed. If dx = 2, then either F is connected with dF = 1 and Y is an affine cone with its vertex removed, or F is compact and connected and dy = 2, where Y is ℂ, the affine quadric Q2, ℙ2 — Q (with Q a quadric curve) or a homogeneous holomorphic * -bundle over an affine cone minus its vertex which is itself an algebraic principal bundle or which admits a two-to-one covering that is.

On Kähler nilmanifolds with top homology in codimension two

2006 ◽  
Vol 74 (1) ◽  
pp. 85-90
Author(s):  
Bruce Gilligan
Keyword(s):  
Lie Group ◽  
Homology Group ◽  
Discrete Subgroup ◽  
Nilpotent Lie Group ◽  
Codimension Two

SupposeGis a connected, complex, nilpotent Lie group and Γ is a discrete subgroup ofGsuch thatG/Γ is Kähler and the top nonvanishing homology group ofG/Γ (with coefficients in ℤ2) is in codimension two or less. We show thatGis then Abelian. We also note that an example from [12] shows that this fails if the top nonvanishing homology is in codimension three.

PROPER ACTIONS ON SOME EXPONENTIAL SOLVABLE HOMOGENEOUS SPACES

2005 ◽  
Vol 16 (09) ◽  
pp. 941-955 ◽  
Author(s):  
ALI BAKLOUTI ◽  
FATMA KHLIF
Keyword(s):  
Maximal Subgroup ◽  
Homogeneous Space ◽  
Lie Group ◽  
Homogeneous Spaces ◽  
Intersection Property ◽  
Nilpotent Lie Group ◽  
Proper Actions ◽  
Simply Connected

Let G be a connected, simply connected nilpotent Lie group, H and K be connected subgroups of G. We show in this paper that the action of K on X = G/H is proper if and only if the triple (G,H,K) has the compact intersection property in both cases where G is at most three-step and where G is special, extending then earlier cases. The result is also proved for exponential homogeneous space on which acts a maximal subgroup.

scholarly journals Poisson Principal Bundles

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.

Basic Forms

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.

scholarly journals On Formality of Some Homogeneous Spaces

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1011
Author(s):  
Aleksy Tralle
Keyword(s):  
Homogeneous Space ◽  
Root System ◽  
Lie Algebra ◽  
Lie Group ◽  
Maximal Torus ◽  
Homogeneous Spaces ◽  
Direct Proof ◽  
Classical Type ◽  
Sasakian Manifolds ◽  
Full Analysis

Let G / H be a homogeneous space of a compact simple classical Lie group G. Assume that the maximal torus T H of H is conjugate to a torus T β whose Lie algebra t β is the kernel of the maximal root β of the root system of the complexified Lie algebra g c . We prove that such homogeneous space is formal. As an application, we give a short direct proof of the formality property of compact homogeneous 3-Sasakian spaces of classical type. This is a complement to the work of Fernández, Muñoz, and Sanchez which contains a full analysis of the formality property of S O ( 3 ) -bundles over the Wolf spaces and the proof of the formality property of homogeneous 3-Sasakian manifolds as a corollary.

scholarly journals Integral geometry under cut loci in compact symmetric spaces

1995 ◽  
Vol 137 ◽  
pp. 33-53 ◽  
Author(s):  
Hiroyuki Tasaki
Keyword(s):  
Invariant Measure ◽  
Lie Group ◽  
Integral Geometry ◽  
Symmetric Spaces ◽  
Homogeneous Spaces ◽  
Integral Invariant ◽  
Identity Component ◽  
Integral Invariants ◽  
Compact Symmetric Spaces ◽  
Group Of Isometries

The theory of integral geometry has mainly treated identities between integral invariants of submanifolds in Riemannian homogeneous spaces like as dμg(g) where M and N are submanifolds in a Riemannian homogeneous spaces of a Lie group G and I(M ∩ gN) is an integral invariant of M ∩ gN. For example Poincaré’s formula is one of typical identities in integral geometry, which is as follows. We denote by M(R2) the identity component of the group of isometries of the plane R2 with a suitable invariant measure μM(R2).

Naturally Reductive Homogeneous Riemannian Manifolds

1985 ◽  
Vol 37 (3) ◽  
pp. 467-487 ◽  
Author(s):  
Carolyn S. Gordon
Keyword(s):  
Homogeneous Space ◽  
Lie Group ◽  
Riemannian Manifolds ◽  
Homogeneous Spaces ◽  
Invariant Metrics ◽  
List Type ◽  
Left Invariant Metrics ◽  
Homogeneous Riemannian Manifolds ◽  
Transitive Subgroup ◽  
Naturally Reductive

The simple algebraic and geometric properties of naturally reductive metrics make them useful as examples in the study of homogeneous Riemannian manifolds. (See for example [2], [3], [15]). The existence and abundance of naturally reductive left-invariant metrics on a Lie group G or homogeneous space G/L reflect the structure of G itself. Such metrics abound on compact groups, exist but are more restricted on noncompact semisimple groups, and are relatively rare on solvable groups. The goals of this paper are(i) to study all naturally reductive homogeneous spaces of G when G is either semisimple of noncompact type or nilpotent and(ii) to give necessary conditions on a Riemannian homogeneous space of an arbitrary Lie group G in order that the metric be naturally reductive with respect to some transitive subgroup of G.

scholarly journals Sphere and projective space of a C*-algebra with a faithful state

2019 ◽  
Vol 52 (1) ◽  
pp. 410-427
Author(s):  
Andrea C. Antunez
Keyword(s):  
Initial Data ◽  
Projective Space ◽  
Banach Spaces ◽  
Lie Group ◽  
Unit Sphere ◽  
Homogeneous Spaces ◽  
Inner Product ◽  
C Algebra ◽  
Faithful State ◽  
Banach Lie Group

AbstractLet 𝒜 be a unital C*-algebra with a faithful state ϕ. We study the geometry of the unit sphere 𝕊ϕ = {x ∈ 𝒜 : ϕ(x*x) = 1} and the projective space ℙϕ = 𝕊ϕ/𝕋. These spaces are shown to be smooth manifolds and homogeneous spaces of the group 𝒰ϕ(𝒜) of isomorphisms acting in 𝒜 which preserve the inner product induced by ϕ, which is a smooth Banach-Lie group. An important role is played by the theory of operators in Banach spaces with two norms, as developed by M.G. Krein and P. Lax. We define a metric in ℙϕ, and prove the existence of minimal geodesics, both with given initial data, and given endpoints.

scholarly journals On orbits of unipotent flows on homogeneous spaces, II

1986 ◽  
Vol 6 (2) ◽  
pp. 167-182 ◽  
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.

Sign in / Sign up

Export Citation Format

Share Document