nLab n-plectic vector space

Contents

Idea
Definition
Related concepts
References

Idea

The generalization of the notion of symplectic vector space from symplectic geometry to n-plectic geometry.

Definition

For $n \in \mathbb{N}$ , an n-plectic vector space is a vector space $V$ (over the real numbers) equipped with an $(n+1)$ -linear skew function

\omega : \wedge^{n+1} V \to \mathbb{R}

such that regarded as a function

V \to \wedge^n V^*

is has trivial kernel.

The concept of Lagrangian subspace also generalizes accordingly.

Let $(V,\Omega)$ be an $n$ -plectic vector space, and $W\subset V$ a subspace. Let $1\leq l \leq n-1$ . We define the $l$ -orthogonal subspace of $W$ as

W^{\perp,l} = \{ v\in V \vert \forall w_1,\cdots,w_l \in W , \imath_{v\wedge w_1\wedge \cdots w_l}\Omega =0\}

Then we say $W$ is $l$ -isotropic, $l$ -coisotropic, and $l$ -lagrangian if $W\subset W^{\perp,l}$ , $W^{\perp,l} \subset W$ , and $W=W^{\perp,l}$ , respectively.

See Section 3 of de Leon et al. 2003 for more.

Related concepts

References

Manuel de León, David Martín de Diego, Aitor Santamaría-Merino. Tulczyjew’s triples and lagrangian submanifolds in classical field theories (2003). (arXiv:math-ph/0302026).

Last revised on April 3, 2024 at 23:11:35. See the history of this page for a list of all contributions to it.