| Title:
|
The Lie groupoid analogue of a symplectic Lie group (English) |
| Author:
|
Pham, David N. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
57 |
| Issue:
|
2 |
| Year:
|
2021 |
| Pages:
|
61-81 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A symplectic Lie group is a Lie group with a left-invariant symplectic form. Its Lie algebra structure is that of a quasi-Frobenius Lie algebra. In this note, we identify the groupoid analogue of a symplectic Lie group. We call the aforementioned structure a $t$-symplectic Lie groupoid; the “$t$" is motivated by the fact that each target fiber of a $t$-symplectic Lie groupoid is a symplectic manifold. For a Lie groupoid $\mathcal{G}\rightrightarrows M$, we show that there is a one-to-one correspondence between quasi-Frobenius Lie algebroid structures on $A\mathcal{G}$ (the associated Lie algebroid) and $t$-symplectic Lie groupoid structures on $\mathcal{G}\rightrightarrows M$. In addition, we also introduce the notion of a symplectic Lie group bundle (SLGB) which is a special case of both a $t$-symplectic Lie groupoid and a Lie group bundle. The basic properties of SLGBs are explored. (English) |
| Keyword:
|
symplectic Lie groups |
| Keyword:
|
Lie groupoids |
| Keyword:
|
symplectic Lie algebroids |
| MSC:
|
22A22 |
| MSC:
|
53D05 |
| idZBL:
|
Zbl 07361066 |
| idMR:
|
MR4306169 |
| DOI:
|
10.5817/AM2021-2-61 |
| . |
| Date available:
|
2021-05-11T14:20:29Z |
| Last updated:
|
2021-11-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148890 |
| . |
| Reference:
|
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| . |
