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        "quote": "> [!example] <a class=\"object-permalink\" href=\"#example-2-10\">例 2.10</a> \n>\n> $n$ 阶<u id=\"concept-2-special-linear-group\" class=\"concept-term\" data-concept-identity=\"urn:uuid:2b8f1e0e-bfcc-4391-ace1-5809a76204e3\" data-concept-route=\"/concept-identities#concept-2b8f1e0e-bfcc-4391-ace1-5809a76204e3\">特殊线性群(special linear group)</u>，又称<u id=\"concept-2-unimodular-group\" class=\"concept-term\" data-concept-identity=\"urn:uuid:f64cefe3-d339-4f0e-ae61-35c7b848b740\" data-concept-route=\"/concept-identities#concept-f64cefe3-d339-4f0e-ae61-35c7b848b740\">幺模群(unimodular group)</u>，记作 $SL(n,\\mathbb{R})$，由行列式为 $1$ 的 $n\\times n$ 实矩阵组成。由<a href=\"#equation-2-5\" class=\"cross-reference\" id=\"xref-03-matrix-groups-3\" data-reference=\"equation-2-5\">式 (2.5)</a>，矩阵乘法的封闭性为\n>\n> $$\n> \\det\\mathsf{A}=\\det\\mathsf{B}=1\\implies\\det(\\mathsf{A}\\mathsf{B})=\\det\\mathsf{A}\\det\\mathsf{B}=1.\n> $$\n>\n> 单位矩阵 $\\mathsf{I}$ 属于 $SL(n,\\mathbb{R})$，因为 $\\det\\mathsf{I}=1$；取逆的封闭性由下式给出：\n>\n> $$\n> \\det\\mathsf{A}=1\\implies\\det\\mathsf{A}^{-1}=\\frac{1}{\\det\\mathsf{A}}=1.\n> $$",
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        "quote": "可以证明，$U(n)$ 中的酉矩阵以及 $\\mathfrak{u}(n)$ 中的矩阵都可由酉矩阵共轭对角化，因此每个 $U(n)$ 矩阵都是李代数 $\\mathfrak{u}(n)$ 中某个矩阵的指数。对 $SO(n)$ 也有相应结论，不过需要将矩阵对角化为 $2\\times2$ 分块。$O(n)$ 则不满足这一性质，因为指数映射无法到达单位元所在连通分支之外的部分。当 $n\\geq2$ 时，[<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/special-linear-group\" data-concept-identity=\"urn:uuid:2b8f1e0e-bfcc-4391-ace1-5809a76204e3\" data-concept-route=\"/concept-identities#concept-2b8f1e0e-bfcc-4391-ace1-5809a76204e3\" id=\"first-c49554d0da92dba324b1-zh\">特殊线性群(special linear group)</u>](../../../../../concepts/linear-algebra/special-linear-group) $SL(n,\\mathbf{R})$ 和 $SL(n,\\mathbf{C})$ 也有不属于指数映射像集的元素，尽管这两个群都是连通的；原文此处把相应的复代数印作 $\\mathfrak{sl}(2,\\mathbf{C})$；本节小结（印刷页 54）则写作 $\\mathfrak{sl}(n,\\mathbf{C})$，因此这里保留原印刷形式并标注为排印疑误。",
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        "quote": "One can show that $U ( n )$ and $\\mathfrak { u } ( n )$ matrices can be diagonalized by conjugation by a unitary matrix and thus show that any $U ( n )$ matrix can be written as an exponential of something in the Lie algebra. The corresponding theorem is also true for $S O ( n )$ but requires looking at diagonalization into 2 by 2 blocks. It is not true for $O ( n )$ (you can’t reach the disconnected component of the identity by exponentiation). It also turns out to not be true for the [<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/special-linear-group\" data-concept-identity=\"urn:uuid:2b8f1e0e-bfcc-4391-ace1-5809a76204e3\" data-concept-route=\"/concept-identities#concept-2b8f1e0e-bfcc-4391-ace1-5809a76204e3\" id=\"first-2a00b0bb9c6def9040ef-en\">special linear groups</u>](../../../../../concepts/linear-algebra/special-linear-group) $SL(n,\\mathbf{R})$ and $SL(n,\\mathbf{C})$ for $n \\geq 2$ (while the groups are connected, they have elements that are not exponentials of any matrix in ${ \\mathfrak { s l } } ( n , \\mathbf { R } )$ or ${ \\mathfrak { s l } } ( 2 , \\mathbf { C } )$ respectively). **Editorial note:** The PDF prints $\\mathfrak{sl}(2,\\mathbf{C})$ here, while the §5.3 summary on printed page 54 gives $\\mathfrak{sl}(n,\\mathbf{C})$; the p.70 notation is likely a typographical error.",
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        "quote": "- [<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/special-linear-group\" data-concept-identity=\"urn:uuid:2b8f1e0e-bfcc-4391-ace1-5809a76204e3\" data-concept-route=\"/concept-identities#concept-2b8f1e0e-bfcc-4391-ace1-5809a76204e3\">特殊线性群(special linear group)</u>](../../../../../concepts/linear-algebra/special-linear-group) $SL(n,\\mathbf{R})$ 和 $SL(n,\\mathbf{C})$ 由行列式为 $1$ 的可逆矩阵构成。它们的李代数 $\\mathfrak{sl}(n,\\mathbf{R})$ 和 $\\mathfrak{sl}(n,\\mathbf{C})$ 分别由所有迹为零的 $n\\times n$ 矩阵构成。",
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        "quote": "- The [<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/special-linear-group\" data-concept-identity=\"urn:uuid:2b8f1e0e-bfcc-4391-ace1-5809a76204e3\" data-concept-route=\"/concept-identities#concept-2b8f1e0e-bfcc-4391-ace1-5809a76204e3\">special linear groups</u>](../../../../../concepts/linear-algebra/special-linear-group) $SL(n,\\mathbf{R})$ and $SL(n,\\mathbf{C})$ are the groups of invertible matrices with determinant one. Their Lie algebras, $\\mathfrak{sl}(n,\\mathbf{R})$ and $\\mathfrak{sl}(n,\\mathbf{C})$, consist of all $n\\times n$ matrices with trace zero.",
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        "quote": "正如群 $S O ( n )$ 有二重覆盖 $S p i n ( n )$，群 $S O ( 3 , 1 )$ 也有二重覆盖 $S p i n ( 3 , 1 )$，我们将证明它可以等同于由行列式为 1 的 2 乘 2 复矩阵构成的群 [<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/special-linear-group\" data-concept-identity=\"urn:uuid:2b8f1e0e-bfcc-4391-ace1-5809a76204e3\" data-concept-route=\"/concept-identities#concept-2b8f1e0e-bfcc-4391-ace1-5809a76204e3\">特殊线性群 $SL(2,\\mathbb C)$(special linear group)</u>](../../../../../concepts/linear-algebra/special-linear-group)。这个群将与 $S O ( 3 , 1 )$ 具有相同的李代数，我们有时会将这两个群中的任一个称为“洛伦兹群”。",
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