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        "quote": "一般的抽象数学观点（我们将在[第5](../chapter-05/index#section-5)章中更详细地讨论）是，表示$\\pi$是流形之间的映射，从李群$U ( 1 )$到李群$GL( n , \\mathbf { C } )$，它将$U ( 1 )$的单位元映到$GL( n , \\mathbf { C } )$的单位元。因此，它有一个微分$\\pi ^ { \\prime }$，这是一个从$U ( 1 )$单位元处的切空间（这里即$i\\mathbf{R}$）到$GL( n , \\mathbf { C } )$单位元处的切空间（即$n$乘$n$复矩阵的空间$M ( n , \\mathbf { C } )$）的线性映射。李群单位元处的切空间称为“[<u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/lie-algebra\" data-concept-identity=\"urn:uuid:dd6db1c0-26e6-4777-a6c3-84c28f8fd354\" data-concept-route=\"/concept-identities#concept-dd6db1c0-26e6-4777-a6c3-84c28f8fd354\" id=\"first-0f378a6753637eff06e0-zh\">李代数(Lie algebra)</u>](../../../../../concepts/lie-theory/lie-algebra)”。在后续章节中，我们将研究许多此类李代数和此类映射$\\pi ^ { \\prime }$的不同例子，其中线性映射$\\pi ^ { \\prime }$常常确定表示$\\pi .$",
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        "quote": "\n> [!definition] 定义 · <span class=\"definition-name\"><a href=\"../../../../../concepts/lie-theory/lie-algebra\"><u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/lie-algebra\" data-concept-identity=\"urn:uuid:dd6db1c0-26e6-4777-a6c3-84c28f8fd354\" data-concept-route=\"/concept-identities#concept-dd6db1c0-26e6-4777-a6c3-84c28f8fd354\">李代数(Lie algebra)</u></a></span>\n> <span id=\"source-block-5012\" class=\"block-anchor\"></span>\n>\n> <span class=\"author-definition-label\" aria-hidden=\"true\">定义（李代数）。</span>设 $G$ 是一个由 $n\\times n$ 可逆矩阵构成的李群。$G$ 的 <u id=\"concept-5-matrix-lie-algebra\" class=\"concept-term\" data-concept-id=\"concept-5-matrix-lie-algebra\" data-concept-identity=\"urn:uuid:dd6db1c0-26e6-4777-a6c3-84c28f8fd354\" data-concept-route=\"/concept-identities#concept-dd6db1c0-26e6-4777-a6c3-84c28f8fd354\">李代数(Lie algebra)</u>（记作 $\\operatorname{Lie}(G)$ 或 $\\mathfrak{g}$）是所有满足对每个 $t\\in\\mathbf{R}$ 都有 $e^{tX}\\in G$ 的 $n\\times n$ 矩阵 $X$ 构成的空间。\n",
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