nLab metalinear group

Idea

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

Definition

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

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

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

$$\array{ Ml(n, \mathbb{R}) &\hookrightarrow& Mp(2n,\mathbb{R}) \\ \downarrow && \downarrow \\ Gl(n,\mathbb{R}) &\hookrightarrow& Sp(2n, \mathbb{R}) } \,.$$

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

Related concepts

• metaplectic group, spin group

• metalinear structure