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

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

such that regarded as a function

$$V\to {\wedge }^n{V}^{*}$$V \to \wedge^n V^*

is has trivial kernel.

Related concepts

• n-plectic manifold

• Heisenberg Lie n-algebra