Abstract
Like the classical Gram-Schmidt theorem for symplectic vector spaces, the sheaf-theoretic version (in which the coefficient algebra sheaf A is appropriately chosen) shows that symplectic A-morphisms on free A-modules of finite rank, defined on a topological space X, induce canonical bases (Theorem 1.1), called symplectic bases. Moreover (Theorem 2.1), if (ℰ, φ) is an A-module (with respect to a ℂ-algebra sheaf A without zero divisors) equipped with an orthosymmetric A-morphism, we show, like in the classical situation, that “componentwise” φ is either symmetric (the (local) geometry is orthogonal) or skew-symmetric (the (local) geometry is symplectic). Theorem 2.1 reduces to the classical case for any free A-module of finite rank.
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Ntumba, P.P. The symplectic Gram-Schmidt theorem and fundamental geometries for A-modules. Czech Math J 62, 265–278 (2012). https://doi.org/10.1007/s10587-012-0012-y
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DOI: https://doi.org/10.1007/s10587-012-0012-y
Keywords
- symplectic A-modules
- symplectic Gram-Schmidt theorem
- symplectic basis
- orthosymmetric A-bilinear forms
- orthogonal/symplectic geometry
- strict integral domain algebra sheaf