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        "quote": "若 $V$ 和 $W$ 是同一域 $\\mathbb{K}$ 上的两个向量空间，则映射 $T : V \\to W$ 称为<u id=\"concept-3-linear-map\" class=\"concept-term\" data-concept-identity=\"urn:uuid:4702877e-2ad4-4875-8a27-ef99339c8553\" data-concept-route=\"/concept-identities#concept-4702877e-2ad4-4875-8a27-ef99339c8553\">线性映射(linear map)</u>，或称从 $V$ 到 $W _ { ; }$ 的<u id=\"concept-3-vector-space-homomorphism\" class=\"concept-term\" data-concept-identity=\"urn:uuid:eba919b4-fdf9-44e7-85e3-f560aa4c7943\" data-concept-route=\"/concept-identities#concept-eba919b4-fdf9-44e7-85e3-f560aa4c7943\">向量空间同态(vector space homomorphism)</u>，如果",
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        "quote": "\n\n> [!definition] 定义 · <span class=\"definition-name\">对偶向量空间(dual vector space)</span>\n> <span id=\"source-block-024\" class=\"block-anchor\"></span>\n>\n> <span class=\"author-definition-label\" aria-hidden=\"true\">定义（[<u id=\"concept-4-dual-vector-space\" class=\"concept-term\" data-concept-id=\"concept-4-dual-vector-space\" data-concept-role=\"definition\" data-concept-reference=\"concepts/linear-algebra/dual-space\" data-concept-identity=\"urn:uuid:07fed5dc-6aa1-4ef6-a20a-58fa52a971f7\" data-concept-route=\"/concept-identities#concept-07fed5dc-6aa1-4ef6-a20a-58fa52a971f7\">对偶向量空间(dual vector space)</u>](../../../../../concepts/linear-algebra/dual-space)）。 </span>对于域 $k$ 上的向量空间 $V$， 对偶向量空间 $V ^ { * }$ 是所有从 $V$ 到 $k$ 的[<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/linear-map\" data-concept-identity=\"urn:uuid:4702877e-2ad4-4875-8a27-ef99339c8553\" data-concept-route=\"/concept-identities#concept-4702877e-2ad4-4875-8a27-ef99339c8553\" id=\"first-4c01999493383d81b50c-zh\">线性映射(linear maps)</u>](../../../../../concepts/linear-algebra/linear-map)构成的向量空间， 即，\n>\n> <span id=\"source-block-025\" class=\"block-anchor\"></span>\n>\n> $$\n> V ^ {*} = \\{l: V \\rightarrow k \\text {   such   that   } l (\\alpha v + \\beta w) = \\alpha l (v) + \\beta l (w) \\}\n> $$\n>\n> <span id=\"source-block-026\" class=\"block-anchor\"></span>\n>\n> 其中 $\\alpha,\\beta\\in k$ 且 $v,w\\in V$。\n",
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        "quote": "令[<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/linear-map\" data-concept-identity=\"urn:uuid:4702877e-2ad4-4875-8a27-ef99339c8553\" data-concept-route=\"/concept-identities#concept-4702877e-2ad4-4875-8a27-ef99339c8553\" id=\"first-7551923c597f76833d41-zh\">线性映射(linear map)</u>](../../../../../concepts/linear-algebra/linear-map.md)作用于 $\\mathcal H=\\mathbf C^2$：",
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        "quote": "Let [<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/linear-map\" data-concept-identity=\"urn:uuid:4702877e-2ad4-4875-8a27-ef99339c8553\" data-concept-route=\"/concept-identities#concept-4702877e-2ad4-4875-8a27-ef99339c8553\" id=\"first-4c01999493383d81b50c-en\">linear maps</u>](../../../../../concepts/linear-algebra/linear-map.md) act on $\\mathcal H=\\mathbf C^2$ as follows:",
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        "quote": "[<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/orthogonal-group\" data-concept-identity=\"urn:uuid:9c0cba6a-c2b4-4c44-b15e-b04e84d239af\" data-concept-route=\"/concept-identities#concept-9c0cba6a-c2b4-4c44-b15e-b04e84d239af\">正交群(orthogonal group)</u>](../../../../../concepts/linear-algebra/orthogonal-group)由保持内积的[<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/linear-map\" data-concept-identity=\"urn:uuid:4702877e-2ad4-4875-8a27-ef99339c8553\" data-concept-route=\"/concept-identities#concept-4702877e-2ad4-4875-8a27-ef99339c8553\">线性变换(linear map)</u>](../../../../../concepts/linear-algebra/linear-map)组成；内积是一种对称双线性形式。现在，我们考察将内积替换为反对称双线性形式 $\\Omega$ 后得到的类似群；$\\Omega$ 决定了相空间的辛几何。定义如下：\n<span id=\"first-ca77d6cb966fe643aba7-zh\" class=\"block-anchor\"></span>",
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        "quote": "给定李群 $G$ 对 $\\mathfrak h$ 的作用 $\\phi_g$，每个 $Z\\in\\mathfrak g$ 都通过一个[<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/linear-map\" data-concept-identity=\"urn:uuid:4702877e-2ad4-4875-8a27-ef99339c8553\" data-concept-route=\"/concept-identities#concept-4702877e-2ad4-4875-8a27-ef99339c8553\">线性映射(linear map)</u>](../../../../../concepts/linear-algebra/linear-map)作用于 $\\mathfrak h$：",
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        "quote": "Given an action $\\phi_g$ of a Lie group $G$ on $\\mathfrak h$, each $Z\\in\\mathfrak g$ acts on $\\mathfrak h$ by a [<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/linear-map\" data-concept-identity=\"urn:uuid:4702877e-2ad4-4875-8a27-ef99339c8553\" data-concept-route=\"/concept-identities#concept-4702877e-2ad4-4875-8a27-ef99339c8553\" id=\"first-7551923c597f76833d41-en\">linear map</u>](../../../../../concepts/linear-algebra/linear-map):",
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