# nLab metalinear group

### Context

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

## Idea

For $n\in ℕ$ The metalinear group is a Lie group that is a ${ℤ}_{2}$-group extension of the general linear group $\mathrm{GL}\left(n,ℝ\right)$.

## Definition

Inside the symplectic group $\mathrm{Sp}\left(2n,ℝ\right)$ sits the general linear group

$\mathrm{Gl}\left(n,ℝ\right)↪\mathrm{Sp}\left(2n,ℝ\right)$Gl(n,\mathbb{R}) \hookrightarrow Sp(2n, \mathbb{R})

as the subgroup that preserves the standard Lagrangian submanifold ${ℝ}^{n}↪{ℝ}^{2n}$. Restriction of the metaplectic group extension along this inclusion defines the metalinear group $\mathrm{Ml}\left(n\right)$

$\begin{array}{ccc}\mathrm{Ml}\left(n,ℝ\right)& ↪& \mathrm{Mp}\left(2n,ℝ\right)\\ ↓& & ↓\\ \mathrm{Gl}\left(n,ℝ\right)& ↪& \mathrm{Sp}\left(2n,ℝ\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ Ml(n, \mathbb{R}) &\hookrightarrow& Mp(2n,\mathbb{R}) \\ \downarrow && \downarrow \\ Gl(n,\mathbb{R}) &\hookrightarrow& Sp(2n, \mathbb{R}) } \,.

Hence a metaplectic structure on a symplectic manifold induces a metalinear structure on its Lagrangian submanifolds.

Revised on July 10, 2012 18:22:04 by Urs Schreiber (134.76.83.9)