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        "quote": "\n\n> [!definition] 定义 · 李超代数\n>\n> 实或复向量空间 $V$ 上的[<u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/lie-superalgebra\" data-concept-identity=\"urn:uuid:3b4db342-d792-4a76-bd33-37c7ca43c704\" data-concept-route=\"/concept-identities#concept-3b4db342-d792-4a76-bd33-37c7ca43c704\" id=\"first-77567078e09f32fab57f-zh\">李超代数(Lie superalgebra)</u>](../../../../../concepts/lie-theory/lie-superalgebra)结构由双线性映射[<u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/lie-superbracket\" data-concept-identity=\"urn:uuid:5c0010c3-e567-45b1-926e-91dd0363f528\" data-concept-route=\"/concept-identities#concept-5c0010c3-e567-45b1-926e-91dd0363f528\" id=\"first-7c533cddbd9d2f52b76c-zh\">李超括号(Lie superbracket)</u>](../../../../../concepts/lie-theory/lie-superbracket) $[\\cdot,\\cdot]_\\pm$ 给出。它在通常的或费米子的生成元 $X,Y,Z$ 上满足\n>\n> $$\n> [X,Y]_\\pm=-(-1)^{|X||Y|}[Y,X]_\\pm\n> $$\n>\n> 以及[<u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/super-jacobi-identity\" data-concept-identity=\"urn:uuid:0bec1410-0ea3-4aa7-a217-f21c439ebd4a\" data-concept-route=\"/concept-identities#concept-0bec1410-0ea3-4aa7-a217-f21c439ebd4a\" id=\"first-e9c0807aa4c7be488beb-zh\">超雅可比恒等式(super-Jacobi identity)</u>](../../../../../concepts/lie-theory/super-jacobi-identity)\n>\n> $$\n> [X,[Y,Z]_\\pm]_\\pm=[[X,Y]_\\pm,Z]_\\pm+(-1)^{|X||Y|}[Y,[X,Z]_\\pm]_\\pm\n> $$\n>\n> 其中，通常生成元的 $|X|=0$，费米子生成元的 $|X|=1$。\n",
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        "quote": "\n\n> [!definition] 定义 · 李超代数\n>\n> 实或复向量空间 $V$ 上的[<u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/lie-superalgebra\" data-concept-identity=\"urn:uuid:3b4db342-d792-4a76-bd33-37c7ca43c704\" data-concept-route=\"/concept-identities#concept-3b4db342-d792-4a76-bd33-37c7ca43c704\" id=\"first-77567078e09f32fab57f-zh\">李超代数(Lie superalgebra)</u>](../../../../../concepts/lie-theory/lie-superalgebra)结构由双线性映射[<u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/lie-superbracket\" data-concept-identity=\"urn:uuid:5c0010c3-e567-45b1-926e-91dd0363f528\" data-concept-route=\"/concept-identities#concept-5c0010c3-e567-45b1-926e-91dd0363f528\" id=\"first-7c533cddbd9d2f52b76c-zh\">李超括号(Lie superbracket)</u>](../../../../../concepts/lie-theory/lie-superbracket) $[\\cdot,\\cdot]_\\pm$ 给出。它在通常的或费米子的生成元 $X,Y,Z$ 上满足\n>\n> $$\n> [X,Y]_\\pm=-(-1)^{|X||Y|}[Y,X]_\\pm\n> $$\n>\n> 以及[<u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/super-jacobi-identity\" data-concept-identity=\"urn:uuid:0bec1410-0ea3-4aa7-a217-f21c439ebd4a\" data-concept-route=\"/concept-identities#concept-0bec1410-0ea3-4aa7-a217-f21c439ebd4a\" id=\"first-e9c0807aa4c7be488beb-zh\">超雅可比恒等式(super-Jacobi identity)</u>](../../../../../concepts/lie-theory/super-jacobi-identity)\n>\n> $$\n> [X,[Y,Z]_\\pm]_\\pm=[[X,Y]_\\pm,Z]_\\pm+(-1)^{|X||Y|}[Y,[X,Z]_\\pm]_\\pm\n> $$\n>\n> 其中，通常生成元的 $|X|=0$，费米子生成元的 $|X|=1$。\n",
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        "quote": "\n\n> [!definition] Definition · Lie superalgebra\n>\n> A [<u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/lie-superalgebra\" data-concept-identity=\"urn:uuid:3b4db342-d792-4a76-bd33-37c7ca43c704\" data-concept-route=\"/concept-identities#concept-3b4db342-d792-4a76-bd33-37c7ca43c704\">Lie superalgebra</u>](../../../../../concepts/lie-theory/lie-superalgebra) structure on a real or complex vector space $V$ is given by a [<u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/lie-superbracket\" data-concept-identity=\"urn:uuid:5c0010c3-e567-45b1-926e-91dd0363f528\" data-concept-route=\"/concept-identities#concept-5c0010c3-e567-45b1-926e-91dd0363f528\" id=\"first-7c533cddbd9d2f52b76c-en\">Lie superbracket</u>](../../../../../concepts/lie-theory/lie-superbracket) $[\\cdot,\\cdot]_\\pm$, a bilinear map satisfying, on usual or fermionic generators $X,Y,Z$,\n>\n> $$\n> [X,Y]_\\pm=-(-1)^{|X||Y|}[Y,X]_\\pm\n> $$\n>\n> and the [<u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/super-jacobi-identity\" data-concept-identity=\"urn:uuid:0bec1410-0ea3-4aa7-a217-f21c439ebd4a\" data-concept-route=\"/concept-identities#concept-0bec1410-0ea3-4aa7-a217-f21c439ebd4a\" id=\"first-e9c0807aa4c7be488beb-en\">super-Jacobi identity</u>](../../../../../concepts/lie-theory/super-jacobi-identity)\n>\n> $$\n> [X,[Y,Z]_\\pm]_\\pm=[[X,Y]_\\pm,Z]_\\pm+(-1)^{|X||Y|}[Y,[X,Z]_\\pm]_\\pm\n> $$\n>\n> where $|X|=0$ for a usual generator and $|X|=1$ for a fermionic generator.\n",
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