Symplectic basis - Imported Knowlpedia Content

Definition

Let $(V,\omega)$ be a $2n$ -dimensional symplectic vector space. A symplectic basis is an ordered basis

(e_1,\ldots,e_n,f_1,\ldots,f_n)

such that, for $1\leq i,j\leq n$ ,

\omega(e_i,e_j)=0,\qquad \omega(f_i,f_j)=0,\qquad \omega(e_i,f_j)=\delta_{ij}.

Alternation then gives $\omega(f_i,e_j)=-\delta_{ij}$ . Thus the definition fixes both the pairings and the ordering of the basis vectors; an arbitrary basis made from individually isotropic vectors need not be symplectic. Such a basis places $\omega$ in its standard normal form.

Matrix form

In a symplectic basis, the matrix of $\omega$ is

J=\begin{pmatrix}0&I_n\\-I_n&0\end{pmatrix}.

Conversely, an ordered basis is symplectic exactly when the matrix of $\omega$ in that basis is $J$ . This converts form-preservation questions into the matrix identity $A^{\mathsf T}JA=J$ .

Adapted decompositions

The spans $E=\operatorname{span}(e_1,\ldots,e_n)$ and $F=\operatorname{span}(f_1,\ldots,f_n)$ are complementary isotropic subspaces. The form pairs $E$ and $F$ perfectly, so $F$ identifies with the dual of $E$ . Each plane $\operatorname{span}(e_i,f_i)$ is symplectic, and $V$ is their symplectically orthogonal direct sum.

Existence and conventions

The symplectic basis theorem guarantees that such a basis exists. Some authors order the vectors as $e_1,f_1,\ldots,e_n,f_n$ , producing a block-diagonal matrix with $2\times2$ symplectic blocks instead of the displayed $J$ . Others reverse the sign of $J$ ; these are ordering conventions, not different symplectic structures.

References

Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001. DOI record. Relevant: §1.1, symplectic bases and standard form.
Victor Guillemin and Shlomo Sternberg, Symplectic Techniques in Physics, Cambridge University Press, 1984. Publisher record. Relevant: Chapter 1, linear symplectic geometry.