
Mathematics 2009
On $\mathcal A$Transvections and Symplectic $\mathcal A$ModulesAbstract: 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{PIDalgebra 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.
