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        "quote": "因此，当 $G=SO(n)$ 时，李代数 $\\mathfrak{so}(n)$ 由满足 $X^T=-X$ 的 $n\\times n$ 实[<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\" id=\"first-d489988c101c19cb1a72-zh\">反对称矩阵(skew-symmetric matrix)</u>](../../../../../concepts/linear-algebra/skew-symmetric-matrix)构成，并带有由矩阵对易子给出的双线性、反对称乘积：",
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        "quote": "在[第 6 章](../chapter-06/index#section-6)中，我们将详细讨论 $n=3$ 的情形。此时，李代数 $\\mathfrak{so}(3)$ 可视为 $\\mathbf{R}^3$，实现为反对称实 $3\\times3$ 矩阵构成的空间，其一组基由 $\\epsilon_{12},\\epsilon_{13},\\epsilon_{23}$ 给出；这些矩阵是[<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(skew-symmetric matrix)</u>](../../../../../concepts/linear-algebra/skew-symmetric-matrix)的例子。",
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        "quote": "这些矩阵构成 $\\mathbf{C}^{n^2}$ 的一个实线性子空间，实维数为 $n^2$。因此，$\\mathfrak{u}(n)$ 是实维数为 $n^2$ 的实向量空间，并非由实矩阵构成的空间；其中的矩阵一般含有复数元素。虽然矩阵元素是复数，但只有斜厄米矩阵的实线性组合仍是斜厄米矩阵（乘以 $i$ 会把斜厄米矩阵变为自伴矩阵）。在这组复斜厄米矩阵中，实矩阵构成一个子空间，形成子李代数 $\\mathfrak{so}(n)$；它由[<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(skew-symmetric matrix)</u>](../../../../../concepts/linear-algebra/skew-symmetric-matrix)构成，也是 $SO(n)\\subset U(n)$ 的李代数。",
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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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}
