nLab
metalinear group

Contents

Idea

For nn \in \mathbb{N} The metalinear group is a Lie group that is a 2\mathbb{Z}_2-group extension of the general linear group GL(n,)GL(n, \mathbb{R}).

Definition

Inside the symplectic group Sp(2n,)Sp(2n, \mathbb{R}) sits the general linear group

Gl(n,)Sp(2n,) Gl(n,\mathbb{R}) \hookrightarrow Sp(2n, \mathbb{R})

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

Ml(n,) Mp(2n,) Gl(n,) Sp(2n,). \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)