scholarly journals HOMOGENEOUS SASAKI AND VAISMAN MANIFOLDS OF UNIMODULAR LIE GROUPS

2019 ◽  
pp. 1-14
Author(s):  
D. ALEKSEEVSKY ◽  
K. HASEGAWA ◽  
Y. KAMISHIMA
Keyword(s):  
Lie Groups ◽  
Special Kind ◽  
Basic Structure ◽  
Complete Classification ◽  
Locally Conformally Kähler ◽  
Vaisman Manifold ◽  
Simply Connected ◽  
Sasaki Manifold ◽  
Vaisman Manifolds

A Vaisman manifold is a special kind of locally conformally Kähler manifold, which is closely related to a Sasaki manifold. In this paper, we show a basic structure theorem of simply connected homogeneous Sasaki and Vaisman manifolds of unimodular Lie groups, up to holomorphic isometry. For the case of unimodular Lie groups, we obtain a complete classification of simply connected Sasaki and Vaisman unimodular Lie groups, up to modification.

Classification of 3-dimensional left-invariant almost paracontact metric structures

2017 ◽  
Vol 17 (3) ◽  
Author(s):  
Giovanni Calvaruso ◽  
Antonella Perrone
Keyword(s):  
Lie Groups ◽  
Three Dimensional ◽  
Complete Classification ◽  
Natural Assumption ◽  
Geometric Properties ◽  
3 Dimensional ◽  
Metric Structures

AbstractWe study left-invariant almost paracontact metric structures on arbitrary three-dimensional Lorentzian Lie groups. We obtain a complete classification and description under a natural assumption, which includes relevant classes as normal and almost para-cosymplectic structures, and we investigate geometric properties of these structures.

scholarly journals Unitary spherical representations of Drinfeld doubles

2018 ◽  
Vol 2018 (742) ◽  
pp. 157-186 ◽  
Author(s):  
Yuki Arano
Keyword(s):  
Quantum Group ◽  
Lie Group ◽  
Complete Classification ◽  
Unitary Representations ◽  
Compact Lie Group ◽  
Drinfeld Double ◽  
Quantum Analogue ◽  
Property T ◽  
Simply Connected

Abstract We study irreducible spherical unitary representations of the Drinfeld double of the q-deformation of a connected simply connected compact Lie group, which can be considered as a quantum analogue of the complexification of the Lie group. In the case of \mathrm{SU}_{q}(3) , we give a complete classification of such representations. As an application, we show the Drinfeld double of the quantum group \mathrm{SU}_{q}(2n+1) has property (T), which also implies central property (T) of the dual of \mathrm{SU}_{q}(2n+1) .

scholarly journals Algebraic Families of Harish-Chandra Pairs

2018 ◽  
Vol 2020 (15) ◽  
pp. 4776-4808 ◽  
Author(s):  
Joseph Bernstein ◽  
Nigel Higson ◽  
Eyal Subag
Keyword(s):  
Lie Groups ◽  
Reductive Group ◽  
Complete Classification ◽  
Compact Form ◽  
The Family

Abstract Mathematical physicists have studied degenerations of Lie groups and their representations, which they call contractions. In this paper we study these contractions, and also other families, within the framework of algebraic families of Harish-Chandra modules. We construct a family that incorporates both a real reductive group and its compact form, separate parts of which have been studied individually as contractions. We give a complete classification of generically irreducible families of Harish-Chandra modules in the case of the family associated to $SL(2,\mathbb{R})$.

scholarly journals On the Balmer spectrum for compact Lie groups

2019 ◽  
Vol 156 (1) ◽  
pp. 39-76
Author(s):  
Tobias Barthel ◽  
J. P. C. Greenlees ◽  
Markus Hausmann
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Complete Classification ◽  
Prime Ideals ◽  
Compact Lie Group ◽  
Compact Lie Groups ◽  
Finite Set

We study the Balmer spectrum of the category of finite $G$-spectra for a compact Lie group $G$, extending the work for finite $G$ by Strickland, Balmer–Sanders, Barthel–Hausmann–Naumann–Nikolaus–Noel–Stapleton and others. We give a description of the underlying set of the spectrum and show that the Balmer topology is completely determined by the inclusions between the prime ideals and the topology on the space of closed subgroups of $G$. Using this, we obtain a complete description of this topology for all abelian compact Lie groups and consequently a complete classification of thick tensor ideals. For general compact Lie groups we obtain such a classification away from a finite set of primes $p$.

Adjoint action of a finite loop space. II

1999 ◽  
Vol 129 (4) ◽  
pp. 773-785
Author(s):  
Norio Iwase ◽  
Akira Kono
Keyword(s):  
Lie Groups ◽  
Loop Space ◽  
Adjoint Action ◽  
Theoretic Approach ◽  
Loop Spaces ◽  
Classifying Space ◽  
Simply Connected ◽  
Prime 2 ◽  
Finite Loop

Adjoint actions of compact simply connected Lie groups are studied by Kozima and the second author based on the series of studies on the classification of simple Lie groups and their cohomologies. At odd primes, the first author showed that there is a homotopy theoretic approach that will prove the results of Kozima and the second author for any 1-connected finite loop spaces. In this paper, we use the rationalization of the classifying space to compute the adjoint actions and the cohomology of classifying spaces assuming torsion free hypothesis, at the prime 2. And, by using Browder's work on the Kudo–Araki operations Q1 for homotopy commutative Hopf spaces, we show the converse for general 1-connected finite loop spaces, at the prime 2. This can be done because the inclusion j: G > BAG satisfies the homotopy commutativity for any non-homotopy commutative loop space G.

Left Invariant Einstein–Randers Metrics on Compact Lie Groups

2012 ◽  
Vol 55 (4) ◽  
pp. 870-881 ◽  
Author(s):  
Hui Wang ◽  
Shaoqiang Deng
Keyword(s):  
Lie Groups ◽  
Complete Classification ◽  
Simple Groups ◽  
Compact Lie Group ◽  
Compact Lie Groups ◽  
Navigation Data ◽  
Group Of Isometries ◽  
Randers Metrics ◽  
Naturally Reductive

AbstractIn this paper we study left invariant Einstein–Randers metrics on compact Lie groups. First, we give a method to construct left invariant non-Riemannian Einstein–Randers metrics on a compact Lie group, using the Zermelo navigation data. Then we prove that this gives a complete classification of left invariant Einstein–Randers metrics on compact simple Lie groups with the underlying Riemannian metric naturally reductive. Further, we completely determine the identity component of the group of isometries for this type of metrics on simple groups. Finally, we study some geometric properties of such metrics. In particular, we give the formulae of geodesics and flag curvature of such metrics.

scholarly journals Cohomogeneity one Alexandrov spaces in low dimensions

2020 ◽  
Vol 58 (2) ◽  
pp. 109-146
Author(s):  
Fernando Galaz-García ◽  
Masoumeh Zarei
Keyword(s):  
Sectional Curvature ◽  
Orbit Space ◽  
Complete Classification ◽  
Cohomogeneity One ◽  
Alexandrov Spaces ◽  
Low Dimensions ◽  
Curvature Bound ◽  
Simply Connected ◽  
Lower Curvature Bound

Abstract Alexandrov spaces are complete length spaces with a lower curvature bound in the triangle comparison sense. When they are equipped with an effective isometric action of a compact Lie group with one-dimensional orbit space, they are said to be of cohomogeneity one. Well-known examples include cohomogeneity-one Riemannian manifolds with a uniform lower sectional curvature bound; such spaces are of interest in the context of non-negative and positive sectional curvature. In the present article we classify closed, simply connected cohomogeneity-one Alexandrov spaces in dimensions 5, 6 and 7. This yields, in combination with previous results for manifolds and Alexandrov spaces, a complete classification of closed, simply connected cohomogeneity-one Alexandrov spaces in dimensions at most 7.

On p-groups with a cyclic commutator subgroup

1975 ◽  
Vol 20 (2) ◽  
pp. 178-198 ◽  
Author(s):  
R. J. Miech
Keyword(s):  
Commutator Subgroup ◽  
Metabelian Group ◽  
Special Kind ◽  
Complete Classification ◽  
Isomorphism Problem ◽  
Point Of View ◽  
Defining Relations ◽  
Group A ◽  
Cyclic Commutator

This paper contains the complete classification of the finite p-groups G where p is an odd prime, G is generated by two elements, and the commutator subgroup of G is cyclic. These groups are a special kind of two-generator metabelian group, a class that has been studied by Szekeres (1965). He determined the defining relations of such groups but, as he noted, a “residual isomorphism problem” remains. The cyclic commutator groups are simple when considered from this first point of view; they have a short, easily derived set of defining relations. However, the isomorphism problem is a bit complicated for the defining relations contain nine parameters and each of these parameters might and can be an invariant of the group.

scholarly journals Non-naturally reductive Einstein metrics on normal homogeneous Einstein manifolds

2020 ◽  
pp. 2050079
Author(s):  
Zaili Yan ◽  
Shaoqiang Deng
Keyword(s):  
Lie Groups ◽  
Einstein Metrics ◽  
Complete Classification ◽  
Semisimple Lie Groups ◽  
Killing Form ◽  
Interesting Phenomenon ◽  
Product Metrics ◽  
Coset Spaces ◽  
Naturally Reductive

A quadruple of Lie groups [Formula: see text], where [Formula: see text] is a compact semisimple Lie group, [Formula: see text] are closed subgroups of [Formula: see text], and the related Casimir constants satisfy certain appropriate conditions, is called a basic quadruple. A basic quadruple is called Einstein if the Killing form metrics on the coset spaces [Formula: see text], [Formula: see text] and [Formula: see text] are all Einstein. In this paper, we first give a complete classification of the Einstein basic quadruples. We then show that, except for very few exceptions, given any quadruple [Formula: see text] in our list, we can produce new non-naturally reductive Einstein metrics on the coset space [Formula: see text], by scaling the Killing form metrics along the complement of [Formula: see text] in [Formula: see text] and along the complement of [Formula: see text] in [Formula: see text]. We also show that on some compact semisimple Lie groups, there exist a large number of left invariant non-naturally reductive Einstein metrics which are not product metrics. This discloses a new interesting phenomenon which has not been described in the literature.

scholarly journals Topological Loops with Decomposable Solvable Multiplication Groups

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Ameer Al-Abayechi ◽  
Ágota Figula
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Three Dimensional ◽  
Solvable Lie Groups ◽  
3 Dimensional ◽  
Multiplication Group ◽  
Simply Connected ◽  
Solvable Lie Group

AbstractIn this paper we deal with the class $$\mathcal {C}$$ C of decomposable solvable Lie groups having dimension six. We determine those Lie groups in $$\mathcal {C}$$ C and their subgroups which are the multiplication groups Mult(L) and the inner mapping groups Inn(L) for three-dimensional connected simply connected topological loops L. This result completes the classification of the at most 6-dimensional solvable multiplication Lie groups of the loops L. Moreover, we obtain that every at most 3-dimensional connected topological proper loop having a solvable Lie group of dimension at most six as its multiplication group is centrally nilpotent of class two.

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