Previous |  Up |  Next

Article

Full entry | PDF   (0.5 MB)
Keywords:
symplectic Lie groups; Lie groupoids; symplectic Lie algebroids
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.
References:
[1] Baues, O., Corté, V.: Symplectic Lie groups, I – III. arXiv:1307.1629 [math.DG]. MR 3499032
[2] Bott, R., Tu, L.: Differential Forms in Algebraic Topology. Springer, 1982. MR 0658304 | Zbl 0496.55001
[3] Chari, V., Pressley, A.: Quantum Groups. Cambridge University Press, 1994.
[4] Chevalley, C.: Theory of Lie Groups. Princeton University Press, 1946. MR 0015396
[5] Chu, B.: Symplectic homogeneous spaces. Trans. Amer. Math. Soc. 197 (1974), 145–159. DOI 10.1090/S0002-9947-1974-0342642-7 | MR 0342642
[6] de Leon, M., Marrero, J., Martínez, E.: Lagrangian submanifolds and dynamics on Lie algebroids. J. Phys. A: Math. Gen. 38 (24) (2005), 241–308. DOI 10.1088/0305-4470/38/24/R01 | MR 2147171
[7] Dufour, J., Zung, N.: Poisson Structures and Their Normal Forms. Berkhäuser Verlag, 2005. MR 2178041
[8] Kosmann-Schwarzbach, Y., Mackenzie, K.: Differential operators and actions of Lie algebroids. Quantization, Poisson brackets and beyond, vol. 315, Amer. Math. Soc., Providence, RI, 2002, pp. 213–233. MR 1958838
[9] Lee, J.M.: Introduction to Smooth Manifolds. Springer Verlag, New York, 2003. MR 1930091
[10] Mackenzie, K.: General Theory of Lie Groupoids and Lie Algebroids. London Mathematical Society Lecture Note Series, vol. 213, Cambridge University Press, 2005. MR 2157566 | Zbl 1078.58011
[11] Macknezie, K.: Lie Groupoids and Lie Algebroids in Differential Geometry. London Mathematical Society Lecture Note Series, vol. 124, Cambridge University Press, 1987. MR 0896907
[12] Marle, C.M.: Differential calculus on a Lie algebroid and Poisson manifolds. arXiv:0804.2451v2 [math.DG], June 200. MR 1969436
[13] Marle, C.M.: Calculus on Lie algebroids, Lie groupoids and Poisson manifolds. Dissertationes Math., vol. 457, Polish Academy of Sciences, 2008. DOI 10.4064/dm457-0-1 | MR 2455155
[14] Nest, R., Tsygan, B.: Deformations of symplectic Lie algebroids, deformations of holomorphic symplectic structures, and index theorems. Asian J. Math. 5 (2001), 599–635. DOI 10.4310/AJM.2001.v5.n4.a2 | MR 1913813
[15] Warner, F.: Foundations of differentiable manifolds and Lie groups. Springer Verlag, 1983. MR 0760450
[16] Weinstein, A.: Symplectic groupoids and Poisson manifolds. Bull. Amer. Math. Soc. 16 (1) (1987), 101–104. DOI 10.1090/S0273-0979-1987-15473-5 | MR 0866024

Partner of