# nLab symplectic vector space

### Context

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

## Definition

A vector space $V$ over a field $k$ is symplectic if it is equipped with an exterior 2-form $\omega \in \Lambda^2_k V$ such that $\omega^{\wedge n}=\omega\wedge\omega\wedge\cdots\wedge\omega$ has the maximal rank.

A subspace $W\subset V$ in a symplectic vector space is isotropic if $\omega(v,v) = 0$ for all $v\in W$ and Lagrangean (or lagrangian) if it is maximal isotropic (not proper subspace in any isotropic subspace). See wikipedia.

type of subspace $W$ of inner product spacecondition on orthogonal space $W^\perp$
isotropic subspace$W \subset W^\perp$
coisotropic subspace$W^\perp \subset W$
Lagrangian subspace$W = W^\perp$(for symplectic form)
symplectic space$W \cap W^\perp = \{0\}$(for symplectic form)

## References

• O. T. O’Meara, Symplectic groups, Math. Surveys 16, Amer. Math. Soc. 1978. xi+122 pp.

Revised on January 1, 2015 22:38:14 by Urs Schreiber (127.0.0.1)