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

Title:

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

Creator:

Ntumba, Patrice P.

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:ee4be10a-ec39-409b-95eb-426ff843f263
uuid:ee4be10a-ec39-409b-95eb-426ff843f263

Subject:

symplectic $\mathcal A$-modules, symplectic basis, orthosymmetric $\mathcal {A}$-bilinear forms, orthogonal/symplectic geometry, and strict integral domain algebra sheaf

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

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.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

265-278

Source:

Czechoslovak Mathematical Journal | 2012 Volume:62 | Number:1

Harvested from:

CDK

Metadata only:

false