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        "quote": "我们主要关注 $G$ 为李群的情形，此时 $G$ 是某个维数的[<u class=\"concept-term\" data-concept-reference=\"concepts/differential-geometry/differentiable-manifold\" data-concept-identity=\"urn:uuid:a0553dab-d811-417e-8a79-67ec620792e7\" data-concept-route=\"/concept-identities#concept-a0553dab-d811-417e-8a79-67ec620792e7\" id=\"first-85de52b9f66f87345b81-zh\">可微流形(differentiable manifold)</u>](../../../../../concepts/differential-geometry/differentiable-manifold)。在这种情形下，我们只考虑由[<u class=\"concept-term\" data-concept-reference=\"concepts/differential-geometry/differentiable-map\" data-concept-identity=\"urn:uuid:6ed47c88-27f1-4afd-88e9-df76c8500ef4\" data-concept-route=\"/concept-identities#concept-6ed47c88-27f1-4afd-88e9-df76c8500ef4\" id=\"first-1298f053f1001f2db6be-zh\">可微映射(differentiable map)</u>](../../../../../concepts/differential-geometry/differentiable-map) $\\pi$ 给出的表示。作为向量空间，$GL(n,\\mathbf{C})$ 是所有 $n\\times n$ 复矩阵构成的 $\\mathbf{C}^{n^2}$；要得到 $GL(n,\\mathbf{C})$，需去掉行列式为零的矩阵，也就是所有不可逆矩阵。在 $G$ 上选定局部坐标后，映射 $\\pi:G\\to GL(n,\\mathbf{C})$ 可由 $2n^2$ 个实值函数表示；$G$ 为可微流形这一条件保证了 $\\pi$ 的导数定义良好。我们的重点不是一般情形，而是量子力学中具有核心意义的某些特定李群及其表示 $\\pi$。对这些表示，$\\pi$ 的可微性可以直接看出。",
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