The symplectic Gram-Schmidt theorem and fundamental geometries for $𝒜$-modules

Ntumba, Patrice P.. "The symplectic Gram-Schmidt theorem and fundamental geometries for $\mathcal {A}$-modules." Czechoslovak Mathematical Journal 62.1 (2012): 265-278. <http://eudml.org/doc/246946>.

@article{Ntumba2012,
abstract = {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 $(\mathcal \{E\}, \phi )$ 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” $\phi $ 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.},
author = {Ntumba, Patrice P.},
journal = {Czechoslovak Mathematical Journal},
keywords = {symplectic $\mathcal \{A\}$-modules; symplectic Gram-Schmidt theorem; symplectic basis; orthosymmetric $\mathcal \{A\}$-bilinear forms; orthogonal/symplectic geometry; strict integral domain algebra sheaf; symplectic -module; symplectic Gram-Schmidt theorem; symplectic basis; orthosymmetric -bilinear forms; orthogonal/symplectic geometry},
language = {eng},
number = {1},
pages = {265-278},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The symplectic Gram-Schmidt theorem and fundamental geometries for $\mathcal \{A\}$-modules},
url = {http://eudml.org/doc/246946},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Ntumba, Patrice P.
TI - The symplectic Gram-Schmidt theorem and fundamental geometries for $\mathcal {A}$-modules
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 1
SP - 265
EP - 278
AB - 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 $(\mathcal {E}, \phi )$ 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” $\phi $ 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.
LA - eng
KW - symplectic $\mathcal {A}$-modules; symplectic Gram-Schmidt theorem; symplectic basis; orthosymmetric $\mathcal {A}$-bilinear forms; orthogonal/symplectic geometry; strict integral domain algebra sheaf; symplectic -module; symplectic Gram-Schmidt theorem; symplectic basis; orthosymmetric -bilinear forms; orthogonal/symplectic geometry
UR - http://eudml.org/doc/246946
ER -