scholarly journals On certain classes of $$\mathbf{Sp}(4,\mathbb{R})$$ symmetric $$G_2$$ structures

2020 ◽  
Author(s):  
Paweł Nurowski
Keyword(s):  
Homogeneous Space ◽  
Lie Group ◽  
Real Form ◽  
Intrinsic Torsion

AbstractWe find two different families of $$\mathbf{Sp}(4,\mathbb{R})$$ Sp ( 4 , R ) symmetric $$G_2$$ G 2 structures in seven dimensions. These are $$G_2$$ G 2 structures with $$G_2$$ G 2 being the split real form of the simple exceptional complex Lie group $$G_2$$ G 2 . The first family has $$\tau _2\equiv 0$$ τ 2 ≡ 0 , while the second family has $$\tau _1\equiv \tau _2\equiv 0$$ τ 1 ≡ τ 2 ≡ 0 , where $$\tau _1$$ τ 1 , $$\tau _2$$ τ 2 are the celebrated $$G_2$$ G 2 -invariant parts of the intrinsic torsion of the $$G_2$$ G 2 structure. The families are different in the sense that the first one lives on a homogeneous space $$\mathbf{Sp}(4,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_l$$ Sp ( 4 , R ) / SL ( 2 , R ) l , and the second one lives on a homogeneous space $$\mathbf{Sp}(4,\mathbb{R})/\mathbf{SL}(2,\mathbb{R})_s$$ Sp ( 4 , R ) / SL ( 2 , R ) s . Here $$\mathbf{SL}(2,\mathbb{R})_l$$ SL ( 2 , R ) l is an $$\mathbf{SL}(2,\mathbb{R})$$ SL ( 2 , R ) corresponding to the $$\mathfrak{sl}(2,\mathbb{R})$$ sl ( 2 , R ) related to the long roots in the root diagram of $$\mathfrak{sp}(4,\mathbb{R})$$ sp ( 4 , R ) , and $$\mathbf{SL}(2,\mathbb{R})_s$$ SL ( 2 , R ) s is an $$\mathbf{SL}(2,\mathbb{R})$$ SL ( 2 , R ) corresponding to the $$\mathfrak{sl}(2,\mathbb{R})$$ sl ( 2 , R ) related to the short roots in the root diagram of $$\mathfrak{sp}(4,\mathbb{R})$$ sp ( 4 , R ) .

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.

CRITERION OF PROPER ACTIONS FOR 3-STEP NILPOTENT LIE GROUPS

2007 ◽  
Vol 18 (07) ◽  
pp. 783-795 ◽  
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.

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.

A Classification of Homogeneous Surfaces

1981 ◽  
Vol 33 (5) ◽  
pp. 1097-1110 ◽  
Author(s):  
A. T. Huckleberry ◽  
E. L. Livorni
Keyword(s):  
Automorphism Group ◽  
Homogeneous Space ◽  
Lie Group ◽  
Compact Complex Manifold ◽  
Coset Space ◽  
Left Coset ◽  
Detailed Classification ◽  
Dimensional Ball ◽  
Lie Subgroup

Throughout this paper a surface is a 2-dimensional (not necessarily compact) complex manifold. A surface X is homogeneous if a complex Lie group G of holomorphic transformations acts holomorphically and transitively on it. Concisely, X is homogeneous if it can be identified with the left coset space G/H, where if is a closed complex Lie subgroup of G. We emphasize that the assumption that G is a complex Lie group is an essential part of the definition. For example, the 2-dimensional ball B2 is certainly “homogeneous” in the sense that its automorphism group acts transitively. But it is impossible to realize B2 as a homogeneous space in the above sense. The purpose of this paper is to give a detailed classification of the homogeneous surfaces. We give explicit descriptions of all possibilities.

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.

Corwin–Greenleaf multiplicity functions for complex semisimple symmetric spaces

2015 ◽  
Vol 26 (05) ◽  
pp. 1550039
Author(s):  
Salma Nasrin
Keyword(s):  
Symmetric Space ◽  
Lie Algebras ◽  
Lie Group ◽  
Symmetric Spaces ◽  
Coadjoint Orbit ◽  
Natural Projection ◽  
Real Form ◽  
Single Orbit ◽  
Complex Semisimple ◽  
Complex Symmetric

Let Gℂ be a complex simple Lie group, GU a compact real form, and [Formula: see text] the natural projection between the dual of the Lie algebras. We prove that, for any coadjoint orbit [Formula: see text] of GU, the intersection of [Formula: see text] with a coadjoint orbit [Formula: see text] of Gℂ is either an empty set or a single orbit of GU if [Formula: see text] is isomorphic to a complex symmetric space.

Сurvature-torsion tensor for Cartan connection

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.

scholarly journals On the Discrete Subgroups and Homogeneous Spaces of Nilpotent Lie Groups

1951 ◽  
Vol 2 ◽  
pp. 95-110 ◽  
Author(s):  
Yozô Matsushima
Keyword(s):  
Homogeneous Space ◽  
Compact Space ◽  
Lie Group ◽  
Homogeneous Spaces ◽  
Discrete Subgroup ◽  
Nilpotent Lie Group ◽  
Discrete Subgroups ◽  
Connected Subgroup ◽  
Structure Constants ◽  
Simply Connected

Recently A, Malcev has shown that the homogeneous space of a connected nilpotent Lie group G is the direct product of a compact space and an Euclidean-space and that the compact space of this direct decomposition is also a homogeneous space of a connected subgroup of G. Any compact homogeneous space M of a connected nilpotent Lie group is of the form where is a connected simply connected nilpotent group whose structure constants are rational numbers in a suitable coordinate system and D is a discrete subgroup of G.

scholarly journals A Note on Simple Anti-Commutative Algebras Obtained from Reductive Homogeneous Spaces

1968 ◽  
Vol 31 ◽  
pp. 105-124 ◽  
Author(s):  
Arthur A. Sagle
Keyword(s):  
Homogeneous Space ◽  
Lie Algebra ◽  
Lie Group ◽  
Closed Subgroup ◽  
Homogeneous Spaces ◽  
Commutative Algebras ◽  
Direct Sum ◽  
Reductive Homogeneous Spaces ◽  
Connected Lie Group

LetGbe a connected Lie group andHa closed subgroup, then the homogeneous spaceM = G/His calledreductiveif there exists a decomposition(subspace direct sum) withwhereg(resp.) is the Lie algebra ofG(resp.H); in this case the pair (g,) is called areductive pair.

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