Adjoint Orbits of Semi-Simple Lie Groups and Lagrangian Submanifolds

2016 ◽  
Vol 60 (2) ◽  
pp. 361-385 ◽  
Author(s):  
Elizabeth Gasparim ◽  
Lino Grama ◽  
Luiz A. B. San Martin
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Symplectic Geometry ◽  
Lagrangian Submanifolds ◽  
Simple Lie Groups ◽  
Adjoint Orbits

AbstractWe give various realizations of the adjoint orbits of a semi-simple Lie group and describe their symplectic geometry. We then use these realizations to identify a family of Lagrangian submanifolds of the orbits.

scholarly journals Approximation of nilpotent orbits for simple Lie groups

2021 ◽  
Vol 56 (2) ◽  
pp. 287-327
Author(s):  
Lucas Fresse ◽  
◽  
Salah Mehdi ◽  
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Finite Union ◽  
Nilpotent Orbits ◽  
Simple Lie Groups ◽  
Special Cases ◽  
Adjoint Orbits

We propose a systematic and topological study of limits \(\lim_{\nu\to 0^+}G_\mathbb{R}\cdot(\nu x)\) of continuous families of adjoint orbits for a non-compact simple real Lie group \(G_\mathbb{R}\). This limit is always a finite union of nilpotent orbits. We describe explicitly these nilpotent orbits in terms of Richardson orbits in the case of hyperbolic semisimple elements. We also show that one can approximate minimal nilpotent orbits or even nilpotent orbits by elliptic semisimple orbits. The special cases of \(\mathrm{SL}_n(\mathbb{R})\) and \(\mathrm{SU}(p,q)\) are computed in detail.

scholarly journals Almost paracontact and parahodge structures on manifolds

1985 ◽  
Vol 99 ◽  
pp. 173-187 ◽  
Author(s):  
Soji Kaneyuki ◽  
Floyd L. Williams
Keyword(s):  
Lie Groups ◽  
Geometric Quantization ◽  
Contact Structures ◽  
Simple Lie Groups ◽  
Coset Spaces ◽  
Adjoint Orbits ◽  
Simply Connected

In this paper we study the paracomplex analogues of almost contact structures, and we introduce and study the notion of parahodge structures on manifolds. In particular, we construct new examples of paracomplex manifolds and we find all simply connected parahermitian symmetric coset spaces, which are the adjoint orbits of noncompact simple Lie groups, with parahodge structures induced by the Killing forms. This is done by (i) observing that a version of the results of A. Morimoto [4] on almost contact structures can be formulated and proved for almost paracontact structures, and by (ii) the methods of geometric quantization [3] applied to parahermitian symmetric triples [1] in conjunction with results of [7]. Two of the main results are Theorem 2.5 (which ties together the above structures) and Corollary 3.9.

scholarly journals Singular fibres of the Gelfand–Cetlin system on MANI( n )*

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170423 ◽  
Author(s):  
D. Bouloc ◽  
E. Miranda ◽  
N.T. Zung
Keyword(s):  
Lie Groups ◽  
Homogeneous Spaces ◽  
Lagrangian Submanifolds ◽  
Unitary Groups ◽  
Direct Products ◽  
Compact Lie Groups ◽  
Lie Groupoids ◽  
Finite Dimensional ◽  
Adjoint Orbits ◽  
Free Actions

In this paper, we show that every singular fibre of the Gelfand–Cetlin system on co-adjoint orbits of unitary groups is a smooth isotropic submanifold which is diffeomorphic to a two-stage quotient of a compact Lie group by free actions of two other compact Lie groups. In many cases, these singular fibres can be shown to be homogeneous spaces or even diffeomorphic to compact Lie groups. We also give a combinatorial formula for computing the dimensions of all singular fibres, and give a detailed description of these singular fibres in many cases, including the so-called (multi-)diamond singularities. These (multi-)diamond singular fibres are degenerate for the Gelfand–Cetlin system, but they are Lagrangian submanifolds diffeomorphic to direct products of special unitary groups and tori. Our methods of study are based on different ideas involving complex ellipsoids, Lie groupoids and also general ideas coming from the theory of singularities of integrable Hamiltonian systems. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

scholarly journals MULTIPLE HAMILTONIAN STRUCTURE OF BOGOYAVLENSKY–TODA LATTICES

2004 ◽  
Vol 16 (02) ◽  
pp. 175-241 ◽  
Author(s):  
PANTELIS A. DAMIANOU
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Hamiltonian Structure ◽  
Poisson Brackets ◽  
Noether Symmetries ◽  
Recursion Operators ◽  
Simple Lie Groups ◽  
Interesting Connection ◽  
Group Symmetries ◽  
Toda Lattices

This paper is mainly a review of the multi-Hamiltonian nature of Toda and generalized Toda lattices corresponding to the classical simple Lie groups but it includes also some new results. The areas investigated include master symmetries, recursion operators, higher Poisson brackets, invariants and group symmetries for the systems. In addition to the positive hierarchy we also consider the negative hierarchy which is crucial in establishing the bi-Hamiltonian structure for each particular simple Lie group. Finally, we include some results on point and Noether symmetries and an interesting connection with the exponents of simple Lie groups. The case of exceptional simple Lie groups is still an open problem.

scholarly journals Diophantine properties of nilpotent Lie groups

2015 ◽  
Vol 151 (6) ◽  
pp. 1157-1188 ◽  
Author(s):  
Menny Aka ◽  
Emmanuel Breuillard ◽  
Lior Rosenzweig ◽  
Nicolas de Saxcé
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Nilpotent Lie Groups ◽  
Finitely Generated ◽  
Derived Length ◽  
Simple Lie Groups ◽  
The Ideal

A finitely generated subgroup ${\rm\Gamma}$ of a real Lie group $G$ is said to be Diophantine if there is ${\it\beta}>0$ such that non-trivial elements in the word ball $B_{{\rm\Gamma}}(n)$ centered at $1\in {\rm\Gamma}$ never approach the identity of $G$ closer than $|B_{{\rm\Gamma}}(n)|^{-{\it\beta}}$. A Lie group $G$ is said to be Diophantine if for every $k\geqslant 1$ a random $k$-tuple in $G$ generates a Diophantine subgroup. Semi-simple Lie groups are conjectured to be Diophantine but very little is proven in this direction. We give a characterization of Diophantine nilpotent Lie groups in terms of the ideal of laws of their Lie algebra. In particular we show that nilpotent Lie groups of class at most $5$, or derived length at most $2$, as well as rational nilpotent Lie groups are Diophantine. We also find that there are non-Diophantine nilpotent and solvable (non-nilpotent) Lie groups.

The mod 2 homology of the space of loops on the exceptional Lie group

1989 ◽  
Vol 112 (3-4) ◽  
pp. 187-202 ◽  
Author(s):  
Akira Kono ◽  
Kazumuto Kozima
Keyword(s):  
Hopf Algebra ◽  
Lie Groups ◽  
Lie Group ◽  
Steenrod Algebra ◽  
Algebra Structure ◽  
Spectral Sequences ◽  
Hopf Algebra Structure ◽  
Simple Lie Groups ◽  
Simply Connected

SynopsisThe Hopf algebra structure of H*(ΩG, F2) and the action of the dual Steenrod algebra are completely and explicitly determined when G isone of the connected, simply connected, exceptional, simple Lie groups. The approach is homological, using connected coverings and spectral sequences.

scholarly journals SAMELSON PRODUCTS IN p-REGULAR SO(2n) AND ITS HOMOTOPY NORMALITY

2017 ◽  
Vol 60 (1) ◽  
pp. 165-174 ◽  
Author(s):  
DAISUKE KISHIMOTO ◽  
MITSUNOBU TSUTAYA
Keyword(s):  
Lie Groups ◽  
Homotopy Type ◽  
Lie Group ◽  
Simple Lie Groups ◽  
Product Of Spheres

AbstractA Lie group is called p-regular if it has the p-local homotopy type of a product of spheres. (Non)triviality of the Samelson products of the inclusions of the factor spheres into p-regular SO(2n(p) is determined, which completes the list of (non)triviality of such Samelson products in p-regular simple Lie groups. As an application, we determine the homotopy normality of the inclusion SO(2n − 1) → SO(2n) in the sense of James at any prime p.

scholarly journals Character estimates of adjoint simple Lie groups

2014 ◽  
Vol 17 (5) ◽  
Author(s):  
Corey Manack
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Simple Lie Groups

AbstractCall a compact, connected, simple Lie group

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