The symplectic Gram-Schmidt theorem and fundamental geometries for $\mathcal{A}$-modules

Ntumba, Patrice P.

Czechoslovak Mathematical Journal, Tome 62 (2012) no. 1, pp. 265-278.

Résumé

Like the classical Gram-Schmidt theorem for symplectic vector spaces, the sheaf-theoretic version (in which the coefficient algebra sheaf $\mathcal{A}$ is appropriately chosen) shows that symplectic $\mathcal{A}$-morphisms on free $\mathcal{A}$-modules of finite rank, defined on a topological space $X$, induce canonical bases (Theorem 1.1), called symplectic bases. Moreover (Theorem 2.1), if $(\mathcal{E},\varphi )$ is an $\mathcal{A}$-module (with respect to a $\mathbb{C}$-algebra sheaf $\mathcal{A}$ without zero divisors) equipped with an orthosymmetric $\mathcal{A}$-morphism, we show, like in the classical situation, that “componentwise” $\varphi$ 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 $\mathcal{A}$-module of finite rank.

Détail

DOI : 10.1007/s10587-012-0012-y

Classification : 16D90, 16S60, 18F20
Mots-clés : symplectic $\mathcal{A}$-modules; symplectic Gram-Schmidt theorem; symplectic basis; orthosymmetric $\mathcal{A}$-bilinear forms; orthogonal/symplectic geometry; strict integral domain algebra sheaf