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        "quote": "> [!exercise] <a class=\"object-permalink\" href=\"#problem-19-17\">习题 19.17</a> \n>\n> 设 $\\mathcal{A}$ 为一个李代数，$X$ 为 $\\mathcal{A}$ 中的任意元素。\n>\n> (a) 证明由 $\\mathbf { a d } _ { X } ( Y ) = [ X , Y ]$ 定义的线性算符 $\\operatorname{ad}_X:\\mathcal{A}\\to\\mathcal{A}$ 是从 $\\mathcal{A}$ 到 $\\mathcal { G L } ( \\mathcal { A } )$ 的李代数同态（称为 <u id=\"concept-19-adjoint-representation\" class=\"concept-term\" data-concept-identity=\"urn:uuid:d6add957-677f-4bff-be23-2cb48b7bbaaa\" data-concept-route=\"/concept-identities#concept-d6add957-677f-4bff-be23-2cb48b7bbaaa\">伴随表示(adjoint representation)</u>）。\n>\n> (b) 对任意李群 $G$，证明由 $C _ { g } ( a ) = g a g ^ { - 1 }$ 定义的每个内自同构 $C _ { g } : G \\to G$（见 2.4 节）都是李群自同构，并且由 $\\operatorname { A d } ( g ) = C _ { g * }$ 定义的映射 $\\operatorname { A d } : G \\to G L ( { \\mathcal { G } } )$ 是李群同态。\n>\n> (c) 证明 $\\mathbf { A d } _ { * } = \\mathbf { a d } .$",
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        "quote": "> Let $\\mathcal{A}$ be a Lie algebra, and $X$ any element of $\\mathcal{A}$.\n>\n> (a) Show that the linear operator $\\operatorname{ad}_X:\\mathcal{A}\\to\\mathcal{A}$ defined by $\\mathbf { a d } _ { X } ( Y ) = [ X , Y ]$ is a Lie algebra homomorphism of $\\mathcal{A}$ into $\\mathcal { G L } ( \\mathcal { A } )$ (called the <u class=\"concept-term\" data-concept-identity=\"urn:uuid:d6add957-677f-4bff-be23-2cb48b7bbaaa\" data-concept-route=\"/concept-identities#concept-d6add957-677f-4bff-be23-2cb48b7bbaaa\" id=\"first-3ad400dd97590b4ed9ed-en\">adjoint representation</u>).\n>\n> (b) For any Lie group $G$ show that each inner automorphism $C _ { g } : G \\to G$ defined by $C _ { g } ( a ) = g a g ^ { - 1 }$ (see <a href=\"../chapter-02/04-homomorphisms-and-isomorphisms#section-2-4\" class=\"cross-reference\" id=\"xref-04-lie-groups-of-transformations-19\" data-reference=\"section-2-4\">Section 2.4</a>) is a Lie group automorphism, and the map $\\operatorname { A d } : G \\to G L ( { \\mathcal { G } } )$ defined by $\\operatorname { A d } ( g ) = C _ { g * }$ is a Lie group homomorphism.\n>\n> (c) Show that $\\mathbf { A d } _ { * } = \\mathbf { a d } .$",
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        "quote": "\n\n> [!definition] 定义 · <span class=\"definition-name\">李代数的伴随表示</span>\n> <span id=\"source-block-5170\" class=\"block-anchor\"></span>\n>\n> $(\\operatorname{ad},\\mathfrak{g})$ 是由下式给出的 <u id=\"concept-5-adjoint-lie-algebra-representation\" class=\"concept-term\" data-concept-id=\"concept-5-adjoint-lie-algebra-representation\" data-concept-identity=\"urn:uuid:d6add957-677f-4bff-be23-2cb48b7bbaaa\" data-concept-route=\"/concept-identities#concept-d6add957-677f-4bff-be23-2cb48b7bbaaa\">伴随李代数表示(adjoint Lie algebra representation)</u>：\n>\n> <span id=\"source-block-5171\" class=\"block-anchor\"></span>\n>\n> $$\n> X\\in\\mathfrak{g}\\mapsto\\operatorname{ad}(X)\n> $$\n>\n> <span id=\"source-block-5172\" class=\"block-anchor\"></span>\n>\n> 其中，$\\operatorname{ad}(X)$ 是从 $\\mathfrak{g}$ 到自身的线性映射，定义为：\n>\n> <span id=\"source-block-5173\" class=\"block-anchor\"></span>\n>\n> $$\n> Y\\mapsto[X,Y]\n> $$\n",
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        "quote": "\n\n> [!definition] 定义 · <span class=\"definition-name\">李代数的伴随表示</span>\n> <span id=\"source-block-5170\" class=\"block-anchor\"></span>\n>\n> $(\\operatorname{ad},\\mathfrak{g})$ 是由下式给出的 <u id=\"concept-5-adjoint-lie-algebra-representation\" class=\"concept-term\" data-concept-id=\"concept-5-adjoint-lie-algebra-representation\" data-concept-identity=\"urn:uuid:d6add957-677f-4bff-be23-2cb48b7bbaaa\" data-concept-route=\"/concept-identities#concept-d6add957-677f-4bff-be23-2cb48b7bbaaa\">伴随李代数表示(adjoint Lie algebra representation)</u>：\n>\n> <span id=\"source-block-5171\" class=\"block-anchor\"></span>\n>\n> $$\n> X\\in\\mathfrak{g}\\mapsto\\operatorname{ad}(X)\n> $$\n>\n> <span id=\"source-block-5172\" class=\"block-anchor\"></span>\n>\n> 其中，$\\operatorname{ad}(X)$ 是从 $\\mathfrak{g}$ 到自身的线性映射，定义为：\n>\n> <span id=\"source-block-5173\" class=\"block-anchor\"></span>\n>\n> $$\n> Y\\mapsto[X,Y]\n> $$\n",
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        "quote": "\n\n> [!definition] Definition · <span class=\"definition-name\">Adjoint Lie algebra representation</span>\n> <span id=\"source-block-5170\" class=\"block-anchor\"></span>\n>\n> $(\\operatorname{ad},\\mathfrak{g})$ is the <u id=\"concept-5-adjoint-lie-algebra-representation\" class=\"concept-term\" data-concept-id=\"concept-5-adjoint-lie-algebra-representation\" data-concept-identity=\"urn:uuid:d6add957-677f-4bff-be23-2cb48b7bbaaa\" data-concept-route=\"/concept-identities#concept-d6add957-677f-4bff-be23-2cb48b7bbaaa\">adjoint Lie algebra representation</u> given by\n>\n> <span id=\"source-block-5171\" class=\"block-anchor\"></span>\n>\n> $$\n> X\\in\\mathfrak{g}\\mapsto\\operatorname{ad}(X)\n> $$\n>\n> <span id=\"source-block-5172\" class=\"block-anchor\"></span>\n>\n> where $\\operatorname{ad}(X)$ is the linear map from $\\mathfrak{g}$ to itself given by\n>\n> <span id=\"source-block-5173\" class=\"block-anchor\"></span>\n>\n> $$\n> Y\\mapsto[X,Y]\n> $$\n",
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            "text": "它是群层面共轭作用的无穷小版本；沿 $e^{tX}Ye^{-tX}$ 对 $t$ 在 $0$ 处求导，得到：\n\n$$\na d (X) (Y) = \\frac {d}{d t} (e ^ {t X} Y e ^ {- t X}) _ {| t = 0} = [ X, Y ]\\tag{5.4}\n$$\n\n",
            "evidence_ids": [
              "e-4cbb4c62bf713e91fb4efd5c"
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          },
          {
            "id": "adjoint-jacobi",
            "text": "$ad$ 保持李括号，因此其同态性质给出雅可比恒等式；这说明该恒等式与伴随作用的存在相联系。",
            "evidence_ids": [
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        "author_views": [
          {
            "id": "woit-adjoint-representation",
            "book_id": "woit-qtgr",
            "edition_id": "2025-10-20",
            "summary": "作者先说明一般李群表示在单位元处求导会给出李代数表示，再对群的伴随表示求导，得到 $ad(X)(Y)=[X,Y]$；随后由表示的同态性质解释雅可比恒等式。",
            "conditions": "李代数按实向量空间处理；该节也提醒，李代数表示不一定来自对应李群的表示。",
            "notation": "群的伴随表示写作 $Ad$，对应的李代数表示写作 $Ad^{\\prime}=ad$。",
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        "reading_suggestions": [
          {
            "id": "adjoint-reading-next",
            "text": "先读第 5.4 节的一般李代数表示定义和群表示求导，再读伴随表示及雅可比恒等式；这样可以看到 $ad$ 的定义如何从共轭作用导出。",
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        "assessment": {
          "composition": "complete",
          "key_material": "available",
          "critical_gaps": [],
          "whole_book_coverage": "not-exhausted"
        },
        "gaps": [
          "已收录材料涵盖定义和推导，但不穷尽本书其他章节中伴随表示的应用。"
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