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Reality of Unipotent elements in Classical Lie Groups

Krishnendu Gongopadhyay, Chandan Maity

Abstract

The aim of this paper is to give a classification of real and strongly real unipotent elements in a classical simple Lie group. To do this, we will introduce an infinitesimal version of the notion of classical reality in a Lie group. This notion has been applied to classify real and strongly real unipotent elements in a classical simple Lie group.

Reality of Unipotent elements in Classical Lie Groups

Abstract

The aim of this paper is to give a classification of real and strongly real unipotent elements in a classical simple Lie group. To do this, we will introduce an infinitesimal version of the notion of classical reality in a Lie group. This notion has been applied to classify real and strongly real unipotent elements in a classical simple Lie group.

Paper Structure

This paper contains 29 sections, 27 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Reality and Strong Reality in Groups
  3. Adjoint Orbits and Reality
  4. Summary of our results
  5. The general linear group
  6. The special linear group
  7. The symplectic group
  8. The orthogonal and unitary groups
  9. The group $\mathrm{SO^{\ast}}(2n)$
  10. Key tools
  11. Structure of the paper
  12. ${\mathrm{Ad}}_G$-reality and Classical Reality
  13. Notation and background
  14. Associated Lie groups and Lie algebras
  15. Classical simple Lie groups and Lie algebras
  16. ...and 14 more sections

Key Result

Lemma \oldthetheorem

Let $G$ be a Lie group with Lie algebra $\mathfrak {g}$. Let $X\in \mathfrak {g}$ be an Ad$_G$-real (resp. strongly Ad$_G$-real) element, then $\exp X$ is real (resp. strongly real) in $G$.

Theorems & Definitions (37)

  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • Corollary \oldthetheorem
  • Remark \oldthetheorem
  • Example \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem: Jacobson-Morozov, cf. CoMc
  • Proposition \oldthetheorem: BCM
  • Proposition \oldthetheorem: BCM
  • ...and 27 more