| Title:
|
The symplectic Gram-Schmidt theorem and fundamental geometries for $\mathcal A$-modules (English) |
| Author:
|
Ntumba, Patrice P. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
62 |
| Issue:
|
1 |
| Year:
|
2012 |
| Pages:
|
265-278 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
symplectic $\mathcal A$-modules |
| Keyword:
|
symplectic Gram-Schmidt theorem |
| Keyword:
|
symplectic basis |
| Keyword:
|
orthosymmetric $\mathcal {A}$-bilinear forms |
| Keyword:
|
orthogonal/symplectic geometry |
| Keyword:
|
strict integral domain algebra sheaf |
| MSC:
|
16D90 |
| MSC:
|
16S60 |
| MSC:
|
18F20 |
| idZBL:
|
Zbl 1249.18008 |
| idMR:
|
MR2899750 |
| DOI:
|
10.1007/s10587-012-0012-y |
| . |
| Date available:
|
2012-03-05T07:31:46Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142056 |
| . |
| Reference:
|
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