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 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 Classification of Symmetry Lie Algebras of the Canonical Geodesic Equations of Five-Dimensional Solvable Lie Algebras

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1354 ◽  
Author(s):  
Hassan Almusawa ◽  
Ryad Ghanam ◽  
Gerard Thompson
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Vector Fields ◽  
Solvable Lie Groups ◽  
Related System ◽  
Geodesic Equations ◽  
Solvable Lie Algebras ◽  
Symmetry Algebras

In this investigation, we present symmetry algebras of the canonical geodesic equations of the indecomposable solvable Lie groups of dimension five, confined to algebras A 5 , 7 a b c to A 18 a . For each algebra, the related system of geodesics is provided. Moreover, a basis for the associated Lie algebra of the symmetry vector fields, as well as the corresponding nonzero brackets, are constructed and categorized.

From Diffeomorphisms to Dark Energy?

2003 ◽  
Vol 18 (33n35) ◽  
pp. 2467-2474 ◽  
Author(s):  
Vincent G. J. Rodgers ◽  
Takeshi Yasuda
Keyword(s):  
Dark Energy ◽  
Lie Algebras ◽  
Virasoro Algebra ◽  
Two Dimensions ◽  
Coadjoint Orbits ◽  
Coadjoint Representation ◽  
Natural Setting ◽  
Affine Lie Algebras ◽  
Yang Mills ◽  
Geometric Action

There are two physical actions that have a natural setting in terms of the coadjoint representation of the algebra of diffeomorphisms and of affine Lie algebras. One is the usual geometric action that comes from coadjoint orbits. The other action lives on the phase space that is transverse to the orbits and are called transverse actions, where Yang-Mills theory in two dimensions is an example. Here we show that the transverse action associated with the Virasoro algebra might contain clues for a theory for dark energy. These actions might also suggests a mechanism for symmetry changing.

scholarly journals The algebraic structure of relativistic wave equations

1978 ◽  
Vol 20 (4) ◽  
pp. 446-486 ◽  
Author(s):  
A. Cant ◽  
C. A. Hurst
Keyword(s):  
Lie Algebras ◽  
Algebraic Structure ◽  
Mass Spectra ◽  
Wave Equations ◽  
Relativistic Wave Equations ◽  
Relativistic Wave ◽  
Partial Classification ◽  
The Family ◽  
Image Position

The algebraic structure of relativistic wave equations of the formis considered. This leads to the problem of finding all Lie algebrasLwhich contain the Lorentz Lie algebraso(3, 1) and also contain a “four-vector” αμa such anLgives rise to a family of wave equations. The simplest possibility is the Bhabha equations whereL≅so(5). Some authors have claimed that this is theonlyone, but it is shown that there are many other possibilities still in accord with physical requirements. Known facts about representations, along with Dynkin's theory of the embeddings of Lie algebras, are used to obtain a partial classification of wave equations. The discrete transformationsC, P, Tare also discussed, along with reality properties. Finally, a simple example of a family of wave equations based onL=sp(12) is considered in detail. Theso(3, 1) content and mass spectra are given for the low order members of the family, and the problem of causality is briefly discussed.

LEFT INVARIANT COMPLEX STRUCTURES ON NILPOTENT SIMPLY CONNECTED INDECOMPOSABLE 6-DIMENSIONAL REAL LIE GROUPS

2007 ◽  
Vol 17 (01) ◽  
pp. 115-139 ◽  
Author(s):  
L. MAGNIN
Keyword(s):  
Automorphism Group ◽  
Normal Form ◽  
Lie Groups ◽  
Lie Algebras ◽  
Lie Group ◽  
Normal Forms ◽  
Equivalence Classes ◽  
Complex Structures ◽  
Simply Connected ◽  
Connected Lie Group

Integrable complex structures on indecomposable 6-dimensional nilpotent real Lie algebras have been computed in a previous paper, along with normal forms for representatives of the various equivalence classes under the action of the automorphism group. Here we go to the connected simply connected Lie group G0 associated to such a Lie algebra 𝔤. For each normal form J of integrable complex structures on 𝔤, we consider the left invariant complex manifold G = (G0, J) associated to G0 and J. We explicitly compute a global holomorphic chart for G and we write down the multiplication in that chart.

KAC-Moody Lie Algebras and the Classification of Nilpotent Lie Algebras of Maximal Rank

1982 ◽  
Vol 34 (6) ◽  
pp. 1215-1239 ◽  
Author(s):  
L. J. Santharoubane
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Dimension ◽  
Maximal Rank ◽  
Killing Form ◽  
Nilpotent Lie Algebra ◽  
Nilpotent Lie Algebras ◽  
Infinite Dimensional ◽  
Dimensional Version

Introduction. The natural problem of determining all the Lie algebras of finite dimension was broken in two parts by Levi's theorem:1) the classification of semi-simple Lie algebras (achieved by Killing and Cartan around 1890)2) the classification of solvable Lie algebras (reduced to the classification of nilpotent Lie algebras by Malcev in 1945 (see [10])).The Killing form is identically equal to zero for a nilpotent Lie algebra but it is non-degenerate for a semi-simple Lie algebra. Therefore there was a huge gap between those two extreme cases. But this gap is only illusory because, as we will prove in this work, a large class of nilpotent Lie algebras is closely related to the Kac-Moody Lie algebras. These last algebras could be viewed as infinite dimensional version of the semisimple Lie algebras.

scholarly journals Some remarks on invariant Poisson quasi-Nijenhuis structures on Lie groups

2019 ◽  
Vol 16 (07) ◽  
pp. 1950097
Author(s):  
Ghorbanali Haghighatdoost ◽  
Zohreh Ravanpak ◽  
Adel Rezaei-Aghdam
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Lie Group ◽  
Baxter Equation ◽  
Complex Structures ◽  
Lie Bialgebra ◽  
Text Structures ◽  
Generalized Complex Structures ◽  
Generalized Complex

We study right-invariant (respectively, left-invariant) Poisson quasi-Nijenhuis structures on a Lie group [Formula: see text] and introduce their infinitesimal counterpart, the so-called r-qn structures on the corresponding Lie algebra [Formula: see text]. We investigate the procedure of the classification of such structures on the Lie algebras and then for clarity of our results we classify, up to a natural equivalence, all [Formula: see text]-[Formula: see text] structures on two types of four-dimensional real Lie algebras. We mention some remarks on the relation between [Formula: see text]-[Formula: see text] structures and the generalized complex structures on the Lie algebras [Formula: see text] and also the solutions of modified Yang–Baxter equation (MYBE) on the double of Lie bialgebra [Formula: see text]. The results are applied to some relevant examples.

scholarly journals Almost Hypercomplex Manifolds with Hermitian-Norden Metrics and 4-dimensional Indecomposable Real Lie Algebras Depending on One Parameter

2020 ◽  
Author(s):  
Hristo Manev
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Smooth Manifolds ◽  
Geometrical Characteristics ◽  
Hypercomplex Structure ◽  
Norden Metrics

We study almost hypercomplex structure with Hermitian-Norden metrics on 4-dimensional Lie groups considered as smooth manifolds. All the basic classes of a classification of 4-dimensional indecomposable real Lie algebras depending on one parameter are investigated. There are studied some geometrical characteristics of the respective almost hypercomplex manifolds with Hermitian-Norden metrics.

scholarly journals Minimal linear representations of the low-dimensional nilpotent Lie algebras

2008 ◽  
Vol 102 (1) ◽  
pp. 17 ◽  
Author(s):  
J. C. Benjumea ◽  
J. Núnez ◽  
A. F. Tenorio
Keyword(s):  
Lie Algebras ◽  
Nilpotent Lie Algebra ◽  
Matrix Representations ◽  
Triangular Matrices ◽  
Nilpotent Lie Algebras ◽  
Linear Representations ◽  
Minimal Dimension ◽  
The Matrix ◽  
Low Dimensional

The main goal of this paper is to compute a minimal matrix representation for each non-isomorphic nilpotent Lie algebra of dimension less than $6$. Indeed, for each of these algebras, we search the natural number $n\in\mathsf{N}\setminus\{1\}$ such that the linear algebra $\mathfrak{g}_n$, formed by all the $n \times n$ complex strictly upper-triangular matrices, contains a representation of this algebra. Besides, we show an algorithmic procedure which computes such a minimal representation by using the Lie algebras $\mathfrak{g}_n$. In this way, a classification of such algebras according to the dimension of their minimal matrix representations is obtained. In this way, we improve some results by Burde related to the value of the minimal dimension of the matrix representations for nilpotent Lie algebras.

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