The Lie groupoid analogue of a symplectic Lie group

Pham, David N.. "The Lie groupoid analogue of a symplectic Lie group." Archivum Mathematicum 057.2 (2021): 61-81. <http://eudml.org/doc/297811>.

@article{Pham2021,
abstract = {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.},
author = {Pham, David N.},
journal = {Archivum Mathematicum},
keywords = {symplectic Lie groups; Lie groupoids; symplectic Lie algebroids},
language = {eng},
number = {2},
pages = {61-81},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {The Lie groupoid analogue of a symplectic Lie group},
url = {http://eudml.org/doc/297811},
volume = {057},
year = {2021},
}

TY - JOUR
AU - Pham, David N.
TI - The Lie groupoid analogue of a symplectic Lie group
JO - Archivum Mathematicum
PY - 2021
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 057
IS - 2
SP - 61
EP - 81
AB - 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.
LA - eng
KW - symplectic Lie groups; Lie groupoids; symplectic Lie algebroids
UR - http://eudml.org/doc/297811
ER -