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.

The Homotopy Type of Lie Semigroups in Semi-Simple Lie Groups

2002 ◽  
Vol 136 (2) ◽  
pp. 151-173 ◽  
Author(s):  
Luiz A. B. San Martin ◽  
Alexandre J. Santana
Keyword(s):  
Lie Groups ◽  
Homotopy Type ◽  
Simple Lie Groups ◽  
Lie Semigroups

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.

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 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

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 Some groups with T1 primitive ideal spaces

1985 ◽  
Vol 38 (1) ◽  
pp. 55-64 ◽  
Author(s):  
A. L. Carey ◽  
W. Moran
Keyword(s):  
Normal Subgroup ◽  
Lie Groups ◽  
Lie Group ◽  
Locally Compact Group ◽  
Primitive Ideal ◽  
Ideal Space ◽  
Solvable Lie Groups ◽  
Locally Compact ◽  
Simply Connected ◽  
Connected Lie Group

AbstractLet G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

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