# 泊松流形

## 定义

M 上一个泊松结构Poisson structure）是一个双线性映射

$\{,\}:C^\infty(M) \times C^\infty(M) \to C^\infty(M),\,$

$\{f,g\}=-\{g,f\},\,$

$\{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f,g\}\}=0,\,$

C(M) 关于第一个变量的导子

$\{fg,h\}=f\{g,h\} + g\{f,h\}$ 对所有 $f,g,h \in C^\infty(M).\,$

$X_g(f) = \{f,g\}\,$

$B_M : \mathrm{T}^* M \to \mathrm{T} M,\,$

## 泊松双向量

$\{f,g\} = \langle \mathrm{d} f \otimes \mathrm{d} g, \eta\rangle ,\,$

$\eta_x=\sum_{i,j=1}^m \eta_{ij}(x) \frac {\partial}{\partial x_i} \otimes \frac {\partial}{\partial x_j}\,$

$\{f,g\}(x)=\sum_{i,j=1}^m \eta_{ij}(x) \frac {\partial f}{\partial x_i} \otimes \frac {\partial g}{\partial x_j}.\,$

## 泊松映射

$\{f_1,f_2\}_N \circ \phi = \{f_1\circ \phi, f_2 \circ \phi\}_M\,$

## 乘积流形

$\{f_1,f_2\}_{M\times N}(x,y) = \{f_1 (x, \cdot), f_2(x, \cdot)\}_N (y) + \{f_1 (\cdot, y), f_2(\cdot, y)\}_M (x)\,$

$f(\cdot,\cdot):M\times N\to\mathbb{R},\,$

$f(x,\cdot):N\to\mathbb{R}\,$

$f(\cdot, y):M\to\mathbb{R}.\,$

## 例子

$\{f_1,f_2\}(x) = \langle \;\left[(df_1)_x, (df_2)_x \right] \,, x \rangle$

$\eta_{ij}(x) = \sum_k c_{ij}^k \langle x, e_k\rangle\,$

## 复结构

$\left(J \otimes J\right)(\eta) = \eta.\,$

## 参考文献

• A. Lichnerowicz, "Les variétès de Poisson et leurs algèbres de Lie associées", J. Diff. Geom. 12 (1977), 253-300.
• A. A. Kirillov, "Local Lie algebras", Russ. Math. Surv. 31 (1976), 55-75.
• V. Guillemin, S. Sternberg, Symplectic Techniques in Physics, Cambridge Univ. Press 1984.
• P. Liberman, C.-M. Marle, Symplectic geometry and analytical mechanics, Reidel 1987.
• K. H. Bhaskara, K. Viswanath, Poisson algebras and Poisson manifolds, Longman 1988, ISBN 0-582-01989-3.
• I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Birkhäuser, 1994. See also the review by Ping Xu in the Bulletin of the AMS.