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        "quote": "复数情况也可作类似讨论：全体 $n\\times n$ 非奇异复矩阵组成群 $GL(n,\\mathbb{C})$。除 $n=1$ 外，这些群均非阿贝尔，因为一般有 $\\mathsf{A}\\mathsf{B}\\ne\\mathsf{B}\\mathsf{A}$。它们并称为$n$ 阶<u id=\"concept-2-general-linear-group\" class=\"concept-term\" data-concept-identity=\"urn:uuid:9a7432a7-4b27-41b4-83d5-d9b6ff0ce028\" data-concept-route=\"/concept-identities#concept-9a7432a7-4b27-41b4-83d5-d9b6ff0ce028\">一般线性群(general linear group)</u>。以矩阵为元素、矩阵乘法为组成法则的子群统称为<u id=\"concept-2-matrix-group\" class=\"concept-term\" data-concept-identity=\"urn:uuid:05f42bf3-d38b-49e0-bed7-1d0452d9ac3d\" data-concept-route=\"/concept-identities#concept-05f42bf3-d38b-49e0-bed7-1d0452d9ac3d\">矩阵群(matrix group)</u>。",
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        "quote": "若群的元素只有有限多个，就称为[<u class=\"concept-term\" data-concept-reference=\"concepts/group-theory/finite-group\" data-concept-identity=\"urn:uuid:91c6b7de-78fd-4b5a-9d1e-2b69ee97ba8e\" data-concept-route=\"/concept-identities#concept-91c6b7de-78fd-4b5a-9d1e-2b69ee97ba8e\" id=\"first-bbdbf409a34a3551c1be-zh\">有限群(finite group)</u>](../../../../../concepts/group-theory/finite-group)。有限群理论及其在量子力学中的应用已相当成熟，但本书大多略过这一主题，转而研究元素个数无限的[<u class=\"concept-term\" data-concept-reference=\"concepts/group-theory/lie-group\" data-concept-identity=\"urn:uuid:8c364498-0e05-4c91-8fd7-1730d6e8656e\" data-concept-route=\"/concept-identities#concept-8c364498-0e05-4c91-8fd7-1730d6e8656e\">李群(Lie group)</u>](../../../../../concepts/group-theory/lie-group)。李群的元素构成某个维数的几何空间；选取局部坐标后，群运算由可微函数给出。本书讨论的大多数李群都是<span id=\"concept-1-matrix-group\" class=\"block-anchor\"></span><u class=\"concept-term\" data-concept-id=\"concept-1-matrix-group\" data-concept-role=\"discussion\" data-concept-identity=\"urn:uuid:05f42bf3-d38b-49e0-bed7-1d0452d9ac3d\" data-concept-route=\"/concept-identities#concept-05f42bf3-d38b-49e0-bed7-1d0452d9ac3d\" id=\"first-fc558e266a7590e7bf39-zh\">矩阵群(matrix group)</u>，也就是由 $n\\times n$ 可逆实矩阵或复矩阵构成的矩阵群的子群。此时群乘法就是矩阵乘法。一个将详细讨论的例子是三维空间中绕某一点旋转所构成的群；这些旋转可以表示为 $3\\times3$ 矩阵，旋转的复合对应矩阵乘法。",
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