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\(A\)-transvections and Witt’s theorem in symplectic \(A\)-modules. (English) Zbl 1254.18013

The paper develops some basic notions and results, originally stated in traditional differential geometry, now in the context of “Abstract Geometric Algebra” [E. Artin, Geometric algebra, Wiley Classics Library. New York etc.: John Wiley & Sons Ltd./Interscience Publishers, Inc. (1988; Zbl 0642.51001)].
The setting is: an algebraized space \((X, \mathcal A)\), where \(\mathcal A\) is a sheaf, over the topological space \(X\), of unital \(\mathbb{C}\)-algebra with some additional property (e.g., the sections over each open have a natural field-like property or are PID). The main notions are of the \(\mathcal A\)-transvection (i.e., a \(\mathcal A\)-hyperplane in a f.g. free \(\mathcal A\)-module) and of \(\mathcal A\)-symplectomorphism. The main results are Witt’s theorem for symplectic vector spaces and the characterization of (singular) symplectomorphisms of symplectic vector spaces of finite (even) dimension.
The main reference is A. Mallios [Geometry of vector sheaves. An axiomatic approach to differential geometry. Volume I: Vector sheaves. General theory. Mathematics and its Applications (Dordrecht). 439. Dordrecht: Kluwer Academic Publishers (1998; Zbl 0904.18001)].

MSC:

18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
13N99 Differential algebra
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
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