A-transvections and Witt’s theorem in symplectic A-modules

Abstract

Building on prior joint work by Mallios and Ntumba, we study transvections (J. Dieudonn´e), a theme already important from the classical theory, in the realm of Abstract Geometric Algebra, referring herewith to symplectic A-modules. A characterization of A-transvections, in terms of A-hyperplanes (Theorem 1.4), is given together with the associated matrix definition (Corollary 1.5). By taking the domain of coefficients A to be a PID-algebra sheaf, we also consider the analogue of a form of the classical Witt’s extension theorem, concerning A-symplectomorphisms defined on appropriate Lagrangian sub-A-modules (Theorem 2.3 and 2.4).