nLab
symplectic vector space

Contents

Definition

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

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

type of subspace WW of inner product spacecondition on orthogonal space W W^\perp
isotropic subspaceWW W \subset W^\perp
coisotropic subspaceW WW^\perp \subset W
Lagrangian subspaceW=W W = W^\perp(for symplectic form)
symplectic spaceWW ={0}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 March 18, 2013 23:48:51 by Urs Schreiber (89.204.138.142)