11、李代数、李群与关联代数详解

李代数、李群与关联代数详解

1. 仿射平面的李代数

在研究仿射平面的李代数时，我们得到了其生成元。这些生成元可以表示为沿积分曲线的向量场，具体如下：
- (L_x = \frac{\partial}{\partial x})
- (L_y = \frac{\partial}{\partial y})
- (L_s = x\frac{\partial}{\partial x} + y\frac{\partial}{\partial y})
- (L_r = -y\frac{\partial}{\partial x} + x\frac{\partial}{\partial y})
- (L_B = x\frac{\partial}{\partial x} - y\frac{\partial}{\partial y})
- (L_b = y\frac{\partial}{\partial x} + x\frac{\partial}{\partial y})

通过对这些无穷小微分生成元取换位子积，我们得到了该李代数的乘法表，如下所示：
| [·, ·] | (L_x) | (L_y) | (L_s) | (L_r) | (L_b) | (L_B) |
| — | — | — | — | — | — | — |
| (L_x) | 0 | 0 | (L_x) | (L_y) | (L_x) | (L_y) |
| (L_y) | 0 | 0 | (L_y) | (-L_x) | (-L_y) | (L_x) |
| (L_s) | (-L_x) | (-L_y) | 0 | 0 | 0 | 0 | <