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        "quote": "则称这个映射是<u id=\"concept-2-homomorphism\" class=\"concept-term\" data-concept-identity=\"urn:uuid:3e749de6-9d6f-43ea-a6de-ddff7c549a00\" data-concept-route=\"/concept-identities#concept-3e749de6-9d6f-43ea-a6de-ddff7c549a00\">同态(homomorphism)</u>",
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        "quote": "\n> [!definition] 定义 · <span class=\"definition-name\"><u class=\"concept-term\" data-concept-id=\"concept-1-representation\" data-concept-identity=\"urn:uuid:1273c312-ea76-4998-85e7-e75280ef3ebb\" data-concept-route=\"/concept-identities#concept-1273c312-ea76-4998-85e7-e75280ef3ebb\">群表示(representation)</u></span>\n> <span id=\"source-block-1085\" class=\"block-anchor\"></span>\n>\n> <span class=\"author-definition-label\" aria-hidden=\"true\">定义（群表示(representation)）。</span>群 $G$ 的表示 $(\\pi,V)$ 是一个[<u class=\"concept-term\" data-concept-reference=\"concepts/group-theory/group-homomorphism\" data-concept-identity=\"urn:uuid:3e749de6-9d6f-43ea-a6de-ddff7c549a00\" data-concept-route=\"/concept-identities#concept-3e749de6-9d6f-43ea-a6de-ddff7c549a00\" id=\"first-da110d60b99705240778-zh\">群同态(group homomorphism)</u>](../../../../../concepts/group-theory/group-homomorphism)：\n>\n> <span id=\"source-block-1086\" class=\"block-anchor\"></span>\n>\n> $$\n> \\pi : g \\in G \\to \\pi (g) \\in GL(V)\n> $$\n>\n> <span id=\"concept-1-general-linear-group\" class=\"block-anchor\"></span>\n>\n> <span id=\"source-block-1087\" class=\"block-anchor\"></span>\n>\n> 其中，$GL(V)$ 是由向量空间 $V$ 到自身的可逆线性映射构成的[<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/general-linear-group\" data-concept-identity=\"urn:uuid:9a7432a7-4b27-41b4-83d5-d9b6ff0ce028\" data-concept-route=\"/concept-identities#concept-9a7432a7-4b27-41b4-83d5-d9b6ff0ce028\" id=\"first-d7a574bbe40adfb64e6d-zh\">一般线性群(general linear group)</u>](../../../../../concepts/linear-algebra/general-linear-group)。\n",
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        "quote": "称映射 $\\pi$ 为[<u class=\"concept-term\" data-concept-reference=\"concepts/group-theory/group-homomorphism\" data-concept-identity=\"urn:uuid:3e749de6-9d6f-43ea-a6de-ddff7c549a00\" data-concept-route=\"/concept-identities#concept-3e749de6-9d6f-43ea-a6de-ddff7c549a00\">群同态(group homomorphism)</u>](../../../../../concepts/group-theory/group-homomorphism)，意思是：",
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        "quote": "我们已经把[<u class=\"concept-term\" data-concept-reference=\"concepts/representation-theory/group-representation\" data-concept-identity=\"urn:uuid:1273c312-ea76-4998-85e7-e75280ef3ebb\" data-concept-route=\"/concept-identities#concept-1273c312-ea76-4998-85e7-e75280ef3ebb\">群表示(group representation)</u>](../../../../../concepts/representation-theory/group-representation)定义为一种[<u class=\"concept-term\" data-concept-reference=\"concepts/group-theory/group-homomorphism\" data-concept-identity=\"urn:uuid:3e749de6-9d6f-43ea-a6de-ddff7c549a00\" data-concept-route=\"/concept-identities#concept-3e749de6-9d6f-43ea-a6de-ddff7c549a00\">群同态(group homomorphism)</u>](../../../../../concepts/group-theory/group-homomorphism)，即保持群乘法的群映射。",
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        "quote": "We have defined a [<u class=\"concept-term\" data-concept-reference=\"concepts/representation-theory/group-representation\" data-concept-identity=\"urn:uuid:1273c312-ea76-4998-85e7-e75280ef3ebb\" data-concept-route=\"/concept-identities#concept-1273c312-ea76-4998-85e7-e75280ef3ebb\" id=\"first-a634d2ae45b7c75364c7-en\">group representation</u>](../../../../../concepts/representation-theory/group-representation) as a [<u class=\"concept-term\" data-concept-reference=\"concepts/group-theory/group-homomorphism\" data-concept-identity=\"urn:uuid:3e749de6-9d6f-43ea-a6de-ddff7c549a00\" data-concept-route=\"/concept-identities#concept-3e749de6-9d6f-43ea-a6de-ddff7c549a00\" id=\"first-2468346c80db9d72875d-en\">homomorphism</u>](../../../../../concepts/group-theory/group-homomorphism) (a map of groups preserving group multiplication).",
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        "quote": "A subtle and remarkable property of the [<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/orthogonal-group\" data-concept-identity=\"urn:uuid:9c0cba6a-c2b4-4c44-b15e-b04e84d239af\" data-concept-route=\"/concept-identities#concept-9c0cba6a-c2b4-4c44-b15e-b04e84d239af\">orthogonal groups</u>](../../../../../concepts/linear-algebra/orthogonal-group) $\\mathrm{SO}(n)$ is that it comes with an associated group, called the [<u id=\"concept-6-spin\" class=\"concept-term\" data-concept-reference=\"concepts/group-theory/spin-group\" data-concept-id=\"concept-6-spin\" data-concept-identity=\"urn:uuid:8adadd91-0a76-4463-a472-62ff66c5bfb3\" data-concept-route=\"/concept-identities#concept-8adadd91-0a76-4463-a472-62ff66c5bfb3\">spin group</u>](../../../../../concepts/group-theory/spin-group) $\\operatorname{Spin}(n)$, such that every element of $\\mathrm{SO}(n)$ corresponds to two distinct elements of $\\operatorname{Spin}(n)$. There is a surjective [<u class=\"concept-term\" data-concept-reference=\"concepts/group-theory/group-homomorphism\" data-concept-identity=\"urn:uuid:3e749de6-9d6f-43ea-a6de-ddff7c549a00\" data-concept-route=\"/concept-identities#concept-3e749de6-9d6f-43ea-a6de-ddff7c549a00\">group homomorphism</u>](../../../../../concepts/group-theory/group-homomorphism)",
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        "quote": "存在一个[<u id=\"concept-6-3-group-homomorphism\" class=\"concept-term\" data-concept-reference=\"concepts/group-theory/group-homomorphism\" data-concept-id=\"concept-6-3-group-homomorphism\" data-concept-identity=\"urn:uuid:3e749de6-9d6f-43ea-a6de-ddff7c549a00\" data-concept-route=\"/concept-identities#concept-3e749de6-9d6f-43ea-a6de-ddff7c549a00\">群同态(group homomorphism)</u>](../../../../../concepts/group-theory/group-homomorphism) $\\Phi:\\operatorname{Spin}(3)\\to\\mathrm{SO}(3)$，它将第一个群映到第二个群，并且是一个二重[<u id=\"concept-6-3-covering-map\" class=\"concept-term\" data-concept-reference=\"concepts/topology/covering-map\" data-concept-id=\"concept-6-3-covering-map\" data-concept-identity=\"urn:uuid:d3626907-41e1-4ef2-bc68-2d2efdec0b42\" data-concept-route=\"/concept-identities#concept-d3626907-41e1-4ef2-bc68-2d2efdec0b42\">覆盖映射(covering map)</u>](../../../../../concepts/topology/covering-map)。其导数 $\\Phi'$ 是两个群的李代数之间的同构。",
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      "quote": "Proof. Both sides are groups according to lemma 3.1. Define a map  $\\varphi : G_1 / \\ker f \\to \\operatorname{im} f$  by  $\\varphi([x]) = f(x)$ . This map is well defined since for  $x' \\in [x]$ , there exists  $h \\in \\ker f$  such that  $x' = x + h$  and  $f(x') = f(x + h) = f(x) + f(h) = f(x)$ . Now we show that  $\\varphi$  is an isomorphism. First,  $\\varphi$  is a homomorphism,",
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