On $\mathcal{A}$-Transvections and Symplectic $\mathcal{A}$-Modules

Abstract

In this paper, building on prior joint work by Mallios and Ntumba, we show that $\mathcal{A}$-\textit{transvections} and \textit{singular symplectic }$\mathcal{A}$-\textit{automorphisms} of symplectic $\mathcal{A}$-modules of finite rank have properties similar to the ones enjoyed by their classical counterparts. The characterization of singular symplectic $\mathcal{A}$-automorphisms of symplectic $\mathcal{A}$-modules of finite rank is grounded on a newly introduced class of pairings of $\mathcal{A}$-modules: the \textit{orthogonally convenient pairings.} We also show that, given a symplectic $\mathcal{A}$-module $\mathcal{E}$ of finite rank, with $\mathcal{A}$ a \textit{PID-algebra sheaf}, any injective $\mathcal{A}$-morphism of a \textit{Lagrangian sub-$\mathcal{A}$-module} $\mathcal{F}$ of $\mathcal{E}$ into $\mathcal{E}$ may be extended to an $\mathcal{A}$-symplectomorphism of $\mathcal{E}$ such that its restriction on $\mathcal{F}$ equals the identity of $\mathcal{F}$. This result also holds in the more general case whereby the underlying free $\mathcal{A}$-module $\mathcal{E}$ is equipped with two symplectic $\mathcal{A}$-structures ${\omega }_0$ and ${\omega }_1$, but with $\mathcal{F}$ being Lagrangian with respect to both ${\omega }_0$ and ${\omega }_1$. The latter is the analog of the classical \textit{Witt's theorem} for symplectic $\mathcal{A}$-modules of finite rank.

30 pages