2018/03/04 by Pham, David N.
#22A22 #53D05 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1803.01289
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\rightrightarrows 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\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.