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 $SL(2,\mathbb{R})$.

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 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 $p$.

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 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.

Foliations Formed by Generic Coadjoint Orbits of a Class of Real Seven-Dimensional Solvable Lie Groups

2021 ◽  
Vol 61 ◽  
pp. 79-104
Author(s):  
Tuyen Nguyen ◽  
◽  
Vu Le
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Topological Classification ◽  
Coadjoint Orbits ◽  
Coadjoint Representation ◽  
Nilpotent Lie Algebra ◽  
The Family ◽  
Measurable Foliation ◽  
Simply Connected

In this paper, we consider exponential, connected and simply connected Lie groups which are corresponding to seven-dimensional Lie algebras such that their nilradical is a five-dimensional nilpotent Lie algebra $\mathfrak{g}_{5,2}$ given in Table~\ref{tab1}. In particular, we give a description of the geometry of the generic orbits in the coadjoint representation of some considered Lie groups. We prove that, for each considered group, the family of the generic coadjoint orbits forms a measurable foliation in the sense of Connes. The topological classification of these foliations is also provided.

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.

scholarly journals Triharmonic Curves in 3-Dimensional Homogeneous Spaces

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
S. Montaldo ◽  
A. Pámpano
Keyword(s):  
Riemannian Manifold ◽  
Homogeneous Spaces ◽  
Arbitrary Dimension ◽  
Complete Classification ◽  
Constant Gaussian Curvature ◽  
Space Forms ◽  
3 Dimensional ◽  
The Family ◽  
Full Classification

AbstractWe first prove that, unlike the biharmonic case, there exist triharmonic curves with nonconstant curvature in a suitable Riemannian manifold of arbitrary dimension. We then give the complete classification of triharmonic curves in surfaces with constant Gaussian curvature. Next, restricting to curves in a 3-dimensional Riemannian manifold, we study the family of triharmonic curves with constant curvature, showing that they are Frenet helices. In the last part, we give the full classification of triharmonic Frenet helices in space forms and in Bianchi–Cartan–Vranceanu spaces.

Three new species, two new genera, and new family Biphragmosagittidae (Сhaetognatha) from Southwest Pacific Ocean

2011 ◽  
Vol 20 (1) ◽  
pp. 161-173
Author(s):  
A.P. Kassatkina
Keyword(s):  
New Species ◽  
Type Species ◽  
Morphological Characters ◽  
The Body ◽  
New Order ◽  
New Genera ◽  
New Family ◽  
The Family ◽  
Three New Species

Resuming published and own data, a revision of classification of Chaetognatha is presented. The family Sagittidae Claus & Grobben, 1905 is given a rank of subclass, Sagittiones, characterised, in particular, by the presence of two pairs of sac-like gelatinous structures or two pairs of fins. Besides the order Aphragmophora Tokioka, 1965, it contains the new order Biphragmosagittiformes ord. nov., which is a unique group of Chaetognatha with an unusual combination of morphological characters: the transverse muscles present in both the trunk and the tail sections of the body; the seminal vesicles simple, without internal complex compartments; the presence of two pairs of lateral fins. The only family assigned to the new order, Biphragmosagittidae fam. nov., contains two genera. Diagnoses of the two new genera, Biphragmosagitta gen. nov. (type species B. tarasovi sp. nov. and B. angusticephala sp. nov.) and Biphragmofastigata gen. nov. (type species B. fastigata sp. nov.), detailed descriptions and pictures of the three new species are presented.

scholarly journals Multiplicative automatic sequences

2021 ◽  
Author(s):  
Jakub Konieczny ◽  
Mariusz Lemańczyk ◽  
Clemens Müllner
Keyword(s):  
Complete Classification ◽  
Automatic Sequences ◽  
Complex Valued

AbstractWe obtain a complete classification of complex-valued sequences which are both multiplicative and automatic.

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