THE LIE GROUPOID ANALOGUE OF A SYMPLECTIC LIE GROUP

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 G paired right arrows M, we show that there is a one-to-one correspondence between quasi-Frobenius Lie algebroid structures on AG (the associated Lie algebroid) and t-symplectic Lie groupoid structures on G paired right arrows 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.