The Lie groupoid analogue of a symplectic Lie group

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 \textit{$t$-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 \textit{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.