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        "quote": "设 $X$ 与 $Y$ 是可微流形 $M$ 的开子集 $U$ 上的光滑向量场。我们将其 <u id=\"concept-15-commutator\" class=\"concept-term\" data-concept-identity=\"urn:uuid:1a15b2f0-5129-43ae-a79b-291a51fb196e\" data-concept-route=\"/concept-identities#concept-1a15b2f0-5129-43ae-a79b-291a51fb196e\">对易子(commutator)</u> 或 <u id=\"concept-15-lie-bracket\" class=\"concept-term\" data-concept-identity=\"urn:uuid:2ec20f01-7d55-4d42-bfe7-cf436d149036\" data-concept-route=\"/concept-identities#concept-2ec20f01-7d55-4d42-bfe7-cf436d149036\">李括号(Lie bracket)</u> [$X$, $Y$] 定义为 $U$ 上由下式定义的向量场",
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        "quote": "Let $X$ and $Y$ be smooth vector fields on an open subset $U$ of a differentiable manifold $M$. We define their <u class=\"concept-term\" data-concept-identity=\"urn:uuid:1a15b2f0-5129-43ae-a79b-291a51fb196e\" data-concept-route=\"/concept-identities#concept-1a15b2f0-5129-43ae-a79b-291a51fb196e\" id=\"first-3cb8369bd67f180c0676-en\">commutator</u> or <u class=\"concept-term\" data-concept-identity=\"urn:uuid:2ec20f01-7d55-4d42-bfe7-cf436d149036\" data-concept-route=\"/concept-identities#concept-2ec20f01-7d55-4d42-bfe7-cf436d149036\" id=\"first-114d1afaafbdde84e335-en\">Lie bracket</u> [$X$, $Y$] as the vector field on $U$ defined by",
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        "quote": "\n> [!definition] 定义 · <span class=\"definition-name\"><a href=\"../../../../../concepts/lie-theory/lie-bracket\"><u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/lie-bracket\" data-concept-identity=\"urn:uuid:2ec20f01-7d55-4d42-bfe7-cf436d149036\" data-concept-route=\"/concept-identities#concept-2ec20f01-7d55-4d42-bfe7-cf436d149036\" id=\"first-51d70695e5ad73ea5e67-zh\">李括号(Lie bracket)</u></a></span>\n> <span id=\"source-block-5039\" class=\"block-anchor\"></span>\n>\n> <span class=\"author-definition-label\" aria-hidden=\"true\">定义（李括号）。</span>在 $\\mathfrak{g}$ 上，<u id=\"concept-5-lie-bracket\" class=\"concept-term\" data-concept-id=\"concept-5-lie-bracket\" data-concept-identity=\"urn:uuid:2ec20f01-7d55-4d42-bfe7-cf436d149036\" data-concept-route=\"/concept-identities#concept-2ec20f01-7d55-4d42-bfe7-cf436d149036\">李括号(Lie bracket)</u>是由矩阵对易子给出的双线性反对称映射\n>\n> <span id=\"source-block-5040\" class=\"block-anchor\"></span>\n>\n> $$\n> [ \\cdot , \\cdot ]: (X, Y) \\in \\mathfrak {g} \\times \\mathfrak {g} \\rightarrow [ X, Y ] = X Y - Y X \\in \\mathfrak {g}\n> $$\n",
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        "quote": "\n> [!definition] 定义 · <span class=\"definition-name\"><a href=\"../../../../../concepts/lie-theory/lie-bracket\"><u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/lie-bracket\" data-concept-identity=\"urn:uuid:2ec20f01-7d55-4d42-bfe7-cf436d149036\" data-concept-route=\"/concept-identities#concept-2ec20f01-7d55-4d42-bfe7-cf436d149036\" id=\"first-51d70695e5ad73ea5e67-zh\">李括号(Lie bracket)</u></a></span>\n> <span id=\"source-block-5039\" class=\"block-anchor\"></span>\n>\n> <span class=\"author-definition-label\" aria-hidden=\"true\">定义（李括号）。</span>在 $\\mathfrak{g}$ 上，<u id=\"concept-5-lie-bracket\" class=\"concept-term\" data-concept-id=\"concept-5-lie-bracket\" data-concept-identity=\"urn:uuid:2ec20f01-7d55-4d42-bfe7-cf436d149036\" data-concept-route=\"/concept-identities#concept-2ec20f01-7d55-4d42-bfe7-cf436d149036\">李括号(Lie bracket)</u>是由矩阵对易子给出的双线性反对称映射\n>\n> <span id=\"source-block-5040\" class=\"block-anchor\"></span>\n>\n> $$\n> [ \\cdot , \\cdot ]: (X, Y) \\in \\mathfrak {g} \\times \\mathfrak {g} \\rightarrow [ X, Y ] = X Y - Y X \\in \\mathfrak {g}\n> $$\n",
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        "quote": "\n\n> [!definition] Definition · <span class=\"definition-name\">Lie bracket</span>\n> <span id=\"source-block-5039\" class=\"block-anchor\"></span>\n>\n> <span class=\"author-definition-label\" aria-hidden=\"true\">Definition (<u class=\"concept-term\" data-concept-id=\"concept-5-lie-bracket\" data-concept-identity=\"urn:uuid:2ec20f01-7d55-4d42-bfe7-cf436d149036\" data-concept-route=\"/concept-identities#concept-2ec20f01-7d55-4d42-bfe7-cf436d149036\">Lie bracket</u>). </span>The Lie bracket operation on $\\mathfrak{g}$ is the bilinear antisymmetric map given by the commutator of matrices\n>\n> <span id=\"source-block-5040\" class=\"block-anchor\"></span>\n>\n> $$\n> [ \\cdot , \\cdot ]: (X, Y) \\in \\mathfrak {g} \\times \\mathfrak {g} \\rightarrow [ X, Y ] = X Y - Y X \\in \\mathfrak {g}\n> $$\n",
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        "quote": "\n\n> [!definition] Definition · <span class=\"definition-name\">Lie bracket</span>\n> <span id=\"source-block-5039\" class=\"block-anchor\"></span>\n>\n> <span class=\"author-definition-label\" aria-hidden=\"true\">Definition (<u class=\"concept-term\" data-concept-id=\"concept-5-lie-bracket\" data-concept-identity=\"urn:uuid:2ec20f01-7d55-4d42-bfe7-cf436d149036\" data-concept-route=\"/concept-identities#concept-2ec20f01-7d55-4d42-bfe7-cf436d149036\">Lie bracket</u>). </span>The Lie bracket operation on $\\mathfrak{g}$ is the bilinear antisymmetric map given by the commutator of matrices\n>\n> <span id=\"source-block-5040\" class=\"block-anchor\"></span>\n>\n> $$\n> [ \\cdot , \\cdot ]: (X, Y) \\in \\mathfrak {g} \\times \\mathfrak {g} \\rightarrow [ X, Y ] = X Y - Y X \\in \\mathfrak {g}\n> $$\n",
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        "quote": "注意，这些对易关系与李代数 $\\mathfrak{su}(2)$ 的基向量 $X_1,X_2,X_3$ 所满足的[方程 3.5](../chapter-03/02-commutation-relations-for-pauli-matrices#equation-3-5) 完全相同，因此 $\\mathfrak{so}(3)$ 与 $\\mathfrak{su}(2)$ 是同构的李代数。二者都是带有相同[<u id=\"concept-6-lie-bracket\" class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/lie-bracket\" data-concept-id=\"concept-6-lie-bracket\" data-concept-identity=\"urn:uuid:2ec20f01-7d55-4d42-bfe7-cf436d149036\" data-concept-route=\"/concept-identities#concept-2ec20f01-7d55-4d42-bfe7-cf436d149036\">李括号(Lie bracket)</u>](../../../../../concepts/lie-theory/lie-bracket)运算的[<u id=\"concept-6-vector-space\" class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/vector-space\" data-concept-id=\"concept-6-vector-space\" data-concept-identity=\"urn:uuid:54e19cc9-1e38-4128-acaa-80499dfd908c\" data-concept-route=\"/concept-identities#concept-54e19cc9-1e38-4128-acaa-80499dfd908c\">向量空间(vector space)</u>](../../../../../concepts/linear-algebra/vector-space) $\\mathbf R^3$。这种运算在另一语境中也很熟悉：它就是标准基向量的[<u id=\"concept-6-cross-product\" class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/vector-cross-product\" data-concept-id=\"concept-6-cross-product\" data-concept-identity=\"urn:uuid:9217da23-ed1d-4016-b7e9-9a1d7c04a546\" data-concept-route=\"/concept-identities#concept-9217da23-ed1d-4016-b7e9-9a1d7c04a546\">叉积(cross product)</u>](../../../../../concepts/linear-algebra/vector-cross-product)：",
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