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"quote": "\n\n> [!definition] 定义（矩映射）\n> 给定 $G$ 对相空间 $M$ 的作用，若有李代数同态\n>\n> $$\n> L\\longmapsto\\mu_L\n> $$\n>\n> 将 $\\mathfrak g$ 映到 $M$ 上的函数，并满足\n>\n> $$\n> X_L=X_{\\mu_L},\n> $$\n>\n> 则称该映射为矩映射。等价地，对于 $M$ 上的函数 $F$，$\\mu_L$ 满足\n>\n> \n>\n> $$\n> \\{\\mu_L,F\\}(m)=-X_LF(m)=\\left.\\frac{d}{dt}F(e^{-tL}\\cdot m)\\right|_{t=0}.\\tag{15.11}\n> $$\n",
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"quote": "\n\n> [!definition] Definition (Moment map)\n> Given an action of $G$ on phase space $M$, a Lie algebra homomorphism\n>\n> $$\n> L\\longmapsto\\mu_L\n> $$\n>\n> from $\\mathfrak g$ to functions on $M$ is called a moment map if\n>\n> $$\n> X_L=X_{\\mu_L}.\n> $$\n>\n> Equivalently, for functions $F$ on $M$, $\\mu_L$ satisfies\n>\n> \n>\n> $$\n> \\{\\mu_L,F\\}(m)=-X_LF(m)=\\left.\\frac{d}{dt}F(e^{-tL}\\cdot m)\\right|_{t=0}.\\tag{15.11}\n> $$\n",
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"text": "在 Woit 第 15.3 节，矩映射把李代数元素 $L$ 对应到相空间 $M$ 上的函数 $\\mu_L$；要求该对应是李代数同态，并且函数所生成的哈密顿向量场与群作用给出的向量场一致：\n\n$$\nX _ {L} = X _ {\\mu_ {L}}\n$$\n\n",
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"summary": "本节先从群作用导出的向量场转向相空间函数，再给出矩映射定义。写成 $\\mu:M\\to\\mathfrak{g}^*$ 时，$\\mu(m)$ 在 $L$ 上的取值就是 $\\mu_L(m)$；作者随后讨论常数选择、哈密顿群作用以及无法满足同态条件时的反常。",
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"notation": "$G$ 是李群，$\\mathfrak{g}$ 是其李代数，$M$ 是相空间，$X_L$ 是群作用诱导的向量场，$\\mu_L$ 是对应的函数。",
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