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        "quote": "这个线性空间 $\\mathfrak{g}\\subset M(n,\\mathbf{C})$ 使我们可以构造一个作用在实向量空间 $\\mathfrak{g}$ 上的特殊表示，称为“<span id=\"concept-5-adjoint-representation\" class=\"block-anchor\"></span>[<u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/adjoint-representation\" data-concept-identity=\"urn:uuid:0c30dd70-94da-4cce-81b0-03aa524fedef\" data-concept-route=\"/concept-identities#concept-0c30dd70-94da-4cce-81b0-03aa524fedef\" id=\"first-039c1d6da9ac735ff8a5-zh\">伴随表示(adjoint representation)</u>](../../../../../concepts/lie-theory/adjoint-representation)”：",
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        "quote": "\n> [!definition] 定义 · <span class=\"definition-name\"><a href=\"../../../../../concepts/lie-theory/adjoint-representation\"><u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/adjoint-representation\" data-concept-identity=\"urn:uuid:0c30dd70-94da-4cce-81b0-03aa524fedef\" data-concept-route=\"/concept-identities#concept-0c30dd70-94da-4cce-81b0-03aa524fedef\">伴随表示(adjoint representation)</u></a></span>\n> <span id=\"source-block-5027\" class=\"block-anchor\"></span>\n>\n> <span class=\"author-definition-label\" aria-hidden=\"true\">定义（伴随表示）。</span><u id=\"concept-5-adjoint-group-representation\" class=\"concept-term\" data-concept-id=\"concept-5-adjoint-group-representation\" data-concept-identity=\"urn:uuid:0c30dd70-94da-4cce-81b0-03aa524fedef\" data-concept-route=\"/concept-identities#concept-0c30dd70-94da-4cce-81b0-03aa524fedef\">伴随表示(adjoint representation)</u> $(\\operatorname{Ad},\\mathfrak{g})$ 是由下列同态给出的表示：\n>\n> <span id=\"source-block-5028\" class=\"block-anchor\"></span>\n>\n> $$\n> \\operatorname{Ad}: g\\in G\\to\\operatorname{Ad}(g)\\in GL(\\mathfrak{g})\n> $$\n>\n> <span id=\"source-block-5029\" class=\"block-anchor\"></span>\n>\n> 其中，$\\operatorname{Ad}(g)$ 对 $X\\in\\mathfrak{g}$ 的作用为\n>\n> <span id=\"source-block-5030\" class=\"block-anchor\"></span>\n>\n> $$\n> (\\operatorname{Ad}(g))(X)=gXg^{-1}\n> $$\n",
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        "quote": "\n> [!definition] 定义 · <span class=\"definition-name\"><a href=\"../../../../../concepts/lie-theory/adjoint-representation\"><u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/adjoint-representation\" data-concept-identity=\"urn:uuid:0c30dd70-94da-4cce-81b0-03aa524fedef\" data-concept-route=\"/concept-identities#concept-0c30dd70-94da-4cce-81b0-03aa524fedef\">伴随表示(adjoint representation)</u></a></span>\n> <span id=\"source-block-5027\" class=\"block-anchor\"></span>\n>\n> <span class=\"author-definition-label\" aria-hidden=\"true\">定义（伴随表示）。</span><u id=\"concept-5-adjoint-group-representation\" class=\"concept-term\" data-concept-id=\"concept-5-adjoint-group-representation\" data-concept-identity=\"urn:uuid:0c30dd70-94da-4cce-81b0-03aa524fedef\" data-concept-route=\"/concept-identities#concept-0c30dd70-94da-4cce-81b0-03aa524fedef\">伴随表示(adjoint representation)</u> $(\\operatorname{Ad},\\mathfrak{g})$ 是由下列同态给出的表示：\n>\n> <span id=\"source-block-5028\" class=\"block-anchor\"></span>\n>\n> $$\n> \\operatorname{Ad}: g\\in G\\to\\operatorname{Ad}(g)\\in GL(\\mathfrak{g})\n> $$\n>\n> <span id=\"source-block-5029\" class=\"block-anchor\"></span>\n>\n> 其中，$\\operatorname{Ad}(g)$ 对 $X\\in\\mathfrak{g}$ 的作用为\n>\n> <span id=\"source-block-5030\" class=\"block-anchor\"></span>\n>\n> $$\n> (\\operatorname{Ad}(g))(X)=gXg^{-1}\n> $$\n",
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        "quote": "\n\n> [!definition] Definition · <span class=\"definition-name\">Adjoint representation</span>\n> <span id=\"source-block-5027\" class=\"block-anchor\"></span>\n>\n> <span class=\"author-definition-label\" aria-hidden=\"true\">Definition (Adjoint representation). </span>The <u id=\"concept-5-adjoint-group-representation\" class=\"concept-term\" data-concept-id=\"concept-5-adjoint-group-representation\" data-concept-identity=\"urn:uuid:0c30dd70-94da-4cce-81b0-03aa524fedef\" data-concept-route=\"/concept-identities#concept-0c30dd70-94da-4cce-81b0-03aa524fedef\">adjoint representation</u> $(Ad,\\mathfrak{g})$ is given by the homomorphism\n>\n> <span id=\"source-block-5028\" class=\"block-anchor\"></span>\n>\n> $$\n> \\operatorname{Ad}: g\\in G\\to\\operatorname{Ad}(g)\\in GL(\\mathfrak{g})\n> $$\n>\n> <span id=\"source-block-5029\" class=\"block-anchor\"></span>\n>\n> where $A d ( g )$ acts on $X \\in { \\mathfrak { g } }$ by\n>\n> <span id=\"source-block-5030\" class=\"block-anchor\"></span>\n>\n> $$\n> (\\operatorname{Ad}(g))(X)=gXg^{-1}\n> $$\n",
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        "quote": "\n\n> [!definition] Definition · <span class=\"definition-name\">Adjoint representation</span>\n> <span id=\"source-block-5027\" class=\"block-anchor\"></span>\n>\n> <span class=\"author-definition-label\" aria-hidden=\"true\">Definition (Adjoint representation). </span>The <u id=\"concept-5-adjoint-group-representation\" class=\"concept-term\" data-concept-id=\"concept-5-adjoint-group-representation\" data-concept-identity=\"urn:uuid:0c30dd70-94da-4cce-81b0-03aa524fedef\" data-concept-route=\"/concept-identities#concept-0c30dd70-94da-4cce-81b0-03aa524fedef\">adjoint representation</u> $(Ad,\\mathfrak{g})$ is given by the homomorphism\n>\n> <span id=\"source-block-5028\" class=\"block-anchor\"></span>\n>\n> $$\n> \\operatorname{Ad}: g\\in G\\to\\operatorname{Ad}(g)\\in GL(\\mathfrak{g})\n> $$\n>\n> <span id=\"source-block-5029\" class=\"block-anchor\"></span>\n>\n> where $A d ( g )$ acts on $X \\in { \\mathfrak { g } }$ by\n>\n> <span id=\"source-block-5030\" class=\"block-anchor\"></span>\n>\n> $$\n> (\\operatorname{Ad}(g))(X)=gXg^{-1}\n> $$\n",
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        "quote": "对于任意李群 $G$，我们已经看到一个特别的表示，即[<u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/adjoint-representation\" data-concept-identity=\"urn:uuid:0c30dd70-94da-4cce-81b0-03aa524fedef\" data-concept-route=\"/concept-identities#concept-0c30dd70-94da-4cce-81b0-03aa524fedef\">伴随表示(adjoint representation)</u>](../../../../../concepts/lie-theory/adjoint-representation) $(\\operatorname{Ad},\\mathfrak{g})$。与之对应的[<u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/lie-algebra-representation\" data-concept-identity=\"urn:uuid:5bad1df2-281a-4c11-8362-128d54fb523a\" data-concept-route=\"/concept-identities#concept-5bad1df2-281a-4c11-8362-128d54fb523a\">李代数表示(Lie algebra representation)</u>](../../../../../concepts/lie-theory/lie-algebra-representation)也称为伴随表示，记作 $(\\operatorname{Ad}',\\mathfrak{g})=(\\operatorname{ad},\\mathfrak{g})$。这是因为",
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        "quote": "For any Lie group $G$, we have seen the distinguished representation, the [<u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/adjoint-representation\" data-concept-identity=\"urn:uuid:0c30dd70-94da-4cce-81b0-03aa524fedef\" data-concept-route=\"/concept-identities#concept-0c30dd70-94da-4cce-81b0-03aa524fedef\">adjoint representation</u>](../../../../../concepts/lie-theory/adjoint-representation) $(\\operatorname{Ad},\\mathfrak{g})$. The corresponding [<u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/lie-algebra-representation\" data-concept-identity=\"urn:uuid:5bad1df2-281a-4c11-8362-128d54fb523a\" data-concept-route=\"/concept-identities#concept-5bad1df2-281a-4c11-8362-128d54fb523a\">Lie algebra representation</u>](../../../../../concepts/lie-theory/lie-algebra-representation) is also called the adjoint representation, written $(\\operatorname{Ad}',\\mathfrak{g})=(\\operatorname{ad},\\mathfrak{g})$. This follows from the fact that",
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        "quote": "列向量上的[<u class=\"concept-term\" data-concept-reference=\"concepts/representation-theory/vector-representation\" data-concept-identity=\"urn:uuid:94ff4c93-90de-45ed-a904-70b63d9c30cf\" data-concept-route=\"/concept-identities#concept-94ff4c93-90de-45ed-a904-70b63d9c30cf\" id=\"first-cd8fe35a8579d4fb05b4-zh\">向量表示</u>](../../../../../concepts/representation-theory/vector-representation) $(\\pi_{\\mathrm{vector}},\\mathbf R^3)$ 与[<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/skew-symmetric-matrix\" data-concept-identity=\"urn:uuid:cbc9ebe8-e9e8-4f3d-a919-cb3725b7f68c\" data-concept-route=\"/concept-identities#concept-cbc9ebe8-e9e8-4f3d-a919-cb3725b7f68c\">反对称矩阵</u>](../../../../../concepts/linear-algebra/skew-symmetric-matrix)上的[<u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/adjoint-representation\" data-concept-identity=\"urn:uuid:0c30dd70-94da-4cce-81b0-03aa524fedef\" data-concept-route=\"/concept-identities#concept-0c30dd70-94da-4cce-81b0-03aa524fedef\" id=\"first-8bf91aaef182ac802e47-zh\">伴随表示</u>](../../../../../concepts/lie-theory/adjoint-representation) $(\\operatorname{Ad},\\mathfrak{so}(3))$ 之间的同构由下式给出",
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        "quote": "The isomorphism between the [<u class=\"concept-term\" data-concept-reference=\"concepts/representation-theory/vector-representation\" data-concept-identity=\"urn:uuid:94ff4c93-90de-45ed-a904-70b63d9c30cf\" data-concept-route=\"/concept-identities#concept-94ff4c93-90de-45ed-a904-70b63d9c30cf\">vector representation</u>](../../../../../concepts/representation-theory/vector-representation) $(\\pi_{\\mathrm{vector}},\\mathbf R^3)$ on column vectors and the [<u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/adjoint-representation\" data-concept-identity=\"urn:uuid:0c30dd70-94da-4cce-81b0-03aa524fedef\" data-concept-route=\"/concept-identities#concept-0c30dd70-94da-4cce-81b0-03aa524fedef\">adjoint representation</u>](../../../../../concepts/lie-theory/adjoint-representation) $(\\operatorname{Ad},\\mathfrak{so}(3))$ on [<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/skew-symmetric-matrix\" data-concept-identity=\"urn:uuid:27cc7d1e-96c7-4375-9829-61dfc96bc2bf\" data-concept-route=\"/concept-identities#concept-27cc7d1e-96c7-4375-9829-61dfc96bc2bf\">skew-symmetric matrices</u>](../../../../../concepts/linear-algebra/skew-symmetric-matrix) is given by",
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      "context": {
        "range": [
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        "quote": "它对这三个群而言都是[<u id=\"concept-7-1-adjoint-representation\" class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/adjoint-representation\" data-concept-id=\"concept-7-1-adjoint-representation\" data-concept-identity=\"urn:uuid:0c30dd70-94da-4cce-81b0-03aa524fedef\" data-concept-route=\"/concept-identities#concept-0c30dd70-94da-4cce-81b0-03aa524fedef\">伴随表示(adjoint representation)</u>](../../../../../concepts/lie-theory/adjoint-representation)；对 $\\mathrm{SO}(3)$ 而言，该表示与[<u id=\"concept-7-1-vector-representation\" class=\"concept-term\" data-concept-reference=\"concepts/representation-theory/vector-representation\" data-concept-id=\"concept-7-1-vector-representation\" data-concept-identity=\"urn:uuid:94ff4c93-90de-45ed-a904-70b63d9c30cf\" data-concept-route=\"/concept-identities#concept-94ff4c93-90de-45ed-a904-70b63d9c30cf\">向量表示(vector representation)</u>](../../../../../concepts/representation-theory/vector-representation)同构。<span id=\"source-7-1-quaternion\" class=\"block-anchor\"></span>",
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      "role": "occurrence",
      "role_basis": {
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        "quote": "This was the [<u id=\"concept-7-1-adjoint-representation\" class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/adjoint-representation\" data-concept-id=\"concept-7-1-adjoint-representation\" data-concept-identity=\"urn:uuid:0c30dd70-94da-4cce-81b0-03aa524fedef\" data-concept-route=\"/concept-identities#concept-0c30dd70-94da-4cce-81b0-03aa524fedef\">adjoint representation</u>](../../../../../concepts/lie-theory/adjoint-representation) for all three groups; for $\\mathrm{SO}(3)$, it is isomorphic to the [<u id=\"concept-7-1-vector-representation\" class=\"concept-term\" data-concept-reference=\"concepts/representation-theory/vector-representation\" data-concept-id=\"concept-7-1-vector-representation\" data-concept-identity=\"urn:uuid:94ff4c93-90de-45ed-a904-70b63d9c30cf\" data-concept-route=\"/concept-identities#concept-94ff4c93-90de-45ed-a904-70b63d9c30cf\">vector representation</u>](../../../../../concepts/representation-theory/vector-representation). <span id=\"source-7-1-quaternion\" class=\"block-anchor\"></span>",
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      "route": "books/woit-qtgr/2025-10-20/zh/chapter-07/03-the-heisenberg-picture",
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      "chapter": "7",
      "section": "7.3",
      "offset": 5163,
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      "source_revision": "5f8a0b2cc9d948646265bae64b9542fa1540f818465f92fddd4b78fdfc0c7c90",
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        "quote": "> [!recall]- 回顾：伴随表示 → [原定义](../chapter-05/01-lie-algebras#concept-5-adjoint-group-representation) <span id=\"source-7-3-adjoint-representation\" class=\"block-anchor\"></span>\n> 在海森堡绘景中，动力学由算符的微分方程描述，而不是由态的微分方程描述。回顾[<u id=\"concept-7-3-adjoint-representation\" class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/adjoint-representation\" data-concept-id=\"concept-7-3-adjoint-representation\" data-concept-identity=\"urn:uuid:0c30dd70-94da-4cce-81b0-03aa524fedef\" data-concept-route=\"/concept-identities#concept-0c30dd70-94da-4cce-81b0-03aa524fedef\">伴随表示(adjoint representation)</u>](../../../../../concepts/lie-theory/adjoint-representation)中的公式（见[方程 (5.1)](../chapter-05/01-lie-algebras#equation-5-1)）：",
        "quote_sha256": "6039cf16fcb74a0ffecd64627ed6f0fde0951c672c7cdd4c7af25b5ba4f4149d",
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      "route": "books/woit-qtgr/2025-10-20/en/chapter-07/03-the-heisenberg-picture",
      "anchor": "concept-7-3-adjoint-representation",
      "chapter": "7",
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      "offset": 5825,
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        "quote": "> [!recall]- Recall: adjoint representation → [original definition](../chapter-05/01-lie-algebras#concept-5-adjoint-group-representation) <span id=\"source-7-3-adjoint-representation\" class=\"block-anchor\"></span>\n> In the Heisenberg picture, the dynamics is governed by a differential equation for the operators rather than the states. Recall the formula from our discussion of the [<u id=\"concept-7-3-adjoint-representation\" class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/adjoint-representation\" data-concept-id=\"concept-7-3-adjoint-representation\" data-concept-identity=\"urn:uuid:0c30dd70-94da-4cce-81b0-03aa524fedef\" data-concept-route=\"/concept-identities#concept-0c30dd70-94da-4cce-81b0-03aa524fedef\">adjoint representation</u>](../../../../../concepts/lie-theory/adjoint-representation) (see [equation (5.1)](../chapter-05/01-lie-algebras#equation-5-1)):",
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        "quote": "这一作用由自同构给出，这些自同构称为内自同构。映射 $\\Phi_g$ 在单位元处的导数，是李代数 $\\mathfrak g$ 上的线性映射 $\\mathrm{Ad}(g)$，称为[<u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/adjoint-representation\" data-concept-identity=\"urn:uuid:0c30dd70-94da-4cce-81b0-03aa524fedef\" data-concept-route=\"/concept-identities#concept-0c30dd70-94da-4cce-81b0-03aa524fedef\">伴随表示(adjoint representation)</u>](../../../../../concepts/lie-theory/adjoint-representation)，见[第5章](../chapter-05/index#section-5)。因此，诱导出的 $\\mathfrak g$ 上的作用为",
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