nLab n-plectic vector space

Idea

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

Definition

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

• n-plectic manifold

• Heisenberg Lie n-algebra

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).