{"id":108809,"date":"2024-03-25T13:54:34","date_gmt":"2024-03-25T13:54:34","guid":{"rendered":"https:\/\/science-hub.click\/E8%28%E6%95%B0%E5%AD%A6%29-%E5%AE%9A%E7%BE%A9\/"},"modified":"2024-03-25T13:54:34","modified_gmt":"2024-03-25T13:54:34","slug":"E8%28%E6%95%B0%E5%AD%A6%29-%E5%AE%9A%E7%BE%A9","status":"publish","type":"post","link":"https:\/\/science-hub.click\/?p=108809","title":{"rendered":"E8 (\u6570\u5b66) – \u5b9a\u7fa9"},"content":{"rendered":"

\u5c0e\u5165<\/h2>
\n\"\"<\/figure>
\u30b4\u30bb\u30c3\u30c8\u30dd\u30ea\u30c8\u30fc\u30d7: \u30eb\u30fc\u30c8\u30b7\u30b9\u30c6\u30e0\u306e 240 \u306e\u30d9\u30af\u30c8\u30eb<\/div><\/div><\/div>

\u6570\u5b66\u3067\u306f\u3001

$$ {E_8\\,} $$<\/div>\u306f\u4f8b\u5916\u7684\u306a\u30bf\u30a4\u30d7\u306e\u6700\u5927\u306e\u8907\u7d20\u30ea\u30fc\u7fa4\u3067\u3059\u3002\u305d\u306e\u30ea\u30fc\u4ee3\u6570\u306f\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u307e\u3059\u3002
$$ {\\mathfrak{e}_8} $$<\/div> \u3002<\/p>

E \u306f\u30e9\u30f3\u30af<\/a><\/span>8\u3001\u6b21\u5143<\/a><\/span>248 \u3067\u3059\u3002\u3053\u308c\u306f\u5358\u7d14\u306b\u63a5\u7d9a\u3055\u308c\u3066\u304a\u308a\u3001\u305d\u306e\u4e2d\u5fc3\u306f\u81ea\u660e\u3067\u3059\u3002<\/p>

E \u69cb\u9020\u306f 1887 \u5e74\u306b\u30ce\u30eb\u30a6\u30a7\u30fc\u306e\u6570\u5b66\u8005<\/a><\/span>\u30bd\u30d5\u30b9 \u30ea\u30fc\u306b\u3088\u3063\u3066\u5bfe\u79f0\u6027\u3092<\/a><\/span>\u7814\u7a76\u3059\u308b\u305f\u3081\u306b\u767a\u898b\u3055\u308c\u307e\u3057\u305f\u304c\u3001\u3053\u308c\u307e\u3067\u8ab0\u3082\u3053\u306e\u6570\u5b66\u7684<\/a><\/span>\u5bfe\u8c61\u304c<\/span>\u7406\u89e3\u3067\u304d\u308b\u3068\u306f\u8003\u3048\u3066\u3044\u306a\u304b\u3063\u305f\u3001\u3068 18 \u4eba\u306e\u6570\u5b66\u8005\u3068\u6570\u5b66\u8005\u304c\u96c6\u307e\u308b\u30a2\u30c8\u30e9\u30b9 \u30aa\u30d6 \u30ea\u30fc\u7fa4\u3068\u8868\u73fe\u30c1\u30fc\u30e0\u306e\u8cac\u4efb\u8005\u30b8\u30a7\u30d5\u30ea\u30fc \u30a2\u30c0\u30e0\u30b9\u306f<\/span>\u8003\u3048\u3066\u3044\u307e\u3059\u3002 Fokko du Cloux \u3084Marc van Leeuwen<\/span>\u306a\u3069\u3001\u4e16\u754c\u4e2d<\/span>\u306e\u30d7\u30ed\u30b0\u30e9\u30de\u30fc\u304c\u53c2\u52a0\u3057\u3066\u3044\u307e\u3059\u3002<\/p>

\n\"<\/figure>

\u5b9f\u4f53<\/h2>

\u8907\u7d20\u6b21\u5143 248 (\u3057\u305f\u304c\u3063\u3066\u5b9f\u6b21\u5143 496) \u306e\u8907\u7d20\u30ea\u30fc\u7fa4<\/a><\/span>E \u306b\u52a0\u3048\u3066\u3001\u3053\u306e\u7fa4\u306b\u306f 3 \u3064\u306e\u5b9f\u6570\u5f62\u304c\u3042\u308a\u3001\u3059\u3079\u3066\u5b9f\u6570 248 \u3067\u3059\u3002\u6700\u3082\u5358\u7d14\u306a\u3082\u306e\u306f\u30b3\u30f3\u30d1\u30af\u30c8\u5f62<\/span>\u3067\u3059\u3002

$$ {E_{8\\left(-248\\right)}} $$<\/div>\u305d\u3057\u3066\u914d\u5099\u3055\u308c\u307e\u3057\u305f<\/span>
$$ {E_{8\\left(8\\right)}} $$<\/div> \uff08\u975e\u30b3\u30f3\u30d1\u30af\u30c8\u6700\u5927\u5024\u307e\u305f\u306f\u82f1\u8a9e\u3067\u306f\u5206\u5272\u3055\u3048\u3042\u308a<\/i>\u307e\u3059\uff09\u305d\u3057\u30663\u756a\u76ee\u304c\u3042\u308a\u307e\u3059\u3002
$$ {E_{8\\left(-24\\right)}} $$<\/div> \u3002<\/p>

\u30b8\u30aa\u30e1\u30c8\u30ea<\/h2>

E \u306e\u30b3\u30f3\u30d1\u30af\u30c8\u306a\u5b9f\u5f62\u5f0f\u306f\u3001\u30aa\u30af\u30c8\u30aa\u30af\u30c8\u30cb\u30aa\u30f3\u5c04\u5f71\u5e73\u9762<\/i>\u3068\u547c\u3070\u308c\u308b\u6b21\u5143 128 \u306e\u30ea\u30fc\u30de\u30f3\u591a\u69d8\u4f53\u306e\u7b49<\/a><\/span>\u9577\u7fa4\u3068\u3057\u3066\u898b\u308b<\/a><\/span>\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u3053\u306e\u540d\u524d\u306f\u3001\u516b\u5143\u6570\u3068\u305d\u308c\u81ea\u4f53\u306e\u30c6\u30f3\u30bd\u30eb\u7a4d<\/a><\/span>\u3068\u3057\u3066\u69cb\u7bc9\u3055\u308c\u308b\u4ee3\u6570<\/a><\/span>\u3092\u4f7f\u7528\u3057\u3066\u69cb\u7bc9\u3067\u304d\u308b\u3068\u3044\u3046\u4e8b\u5b9f\u306b\u7531\u6765\u3057\u3066\u3044\u307e\u3059\u3002\u3053\u306e\u30bf\u30a4\u30d7\u306e\u69cb\u9020\u306f\u3001\u30cf\u30f3\u30b9 \u30d5\u30ed\u30a4\u30c7\u30f3\u30bf\u30fc\u30eb<\/a><\/span>\u3068\u30b8\u30e3\u30c3\u30af \u30c6\u30a3\u30c3\u30c4\u306b\u3088\u3063\u3066\u3001\u9b54\u65b9\u9663<\/a><\/span>\u306e\u69cb\u9020\u306b\u304a\u3044\u3066\u8a73\u7d30\u306b\u5206\u6790\u3055\u308c\u3066\u3044\u307e\u3059\u3002<\/p>

\u5efa\u7269<\/h2>

\u7fa4 E \u306e\u30b3\u30f3\u30d1\u30af\u30c8\u306a\u5f62\u5f0f\u3092\u30ea\u30fc\u4ee3\u6570<\/a><\/span>\u306e\u81ea\u5df1\u540c\u578b\u7fa4\u3068\u3057\u3066\u69cb\u7bc9\u3067\u304d\u308b

$$ {\\mathfrak{e}_8} $$<\/div>\u5bfe\u5fdc\u3057\u3066\u3044\u307e\u3059\u3002\u3053\u306e\u4ee3\u6570\u306f\u3001
$$ {\\mathfrak{so}(16)} $$<\/div>\u6b21\u5143 120 \u306e\u90e8\u5206\u4ee3\u6570\u3068\u3057\u3066\u3001\u3053\u308c\u3092\u4f7f\u7528\u3057\u3066\u968f\u4f34\u8868\u73fe\u3092\u6b21\u306e\u3088\u3046\u306b\u5206\u89e3\u3067\u304d\u307e\u3059\u3002 <\/p>
$$ {\\mathfrak{e}_8 = \\mathfrak{so}(16) \\oplus \\textstyle{S}_{16}^+} $$<\/div><\/dd><\/dl>

\u307e\u305f\u306f

$$ {S_{16}^+} $$<\/div>\u7fa4\u306e\u30de\u30e8\u30e9\u30ca\u30fb\u30ef\u30a4\u30eb\u578b\u306e 2 \u3064\u306e\u30b9\u30d4\u30ce\u30eb\u8868\u73fe\u306e\u3046\u3061\u306e 1 \u3064
$$ {\\operatorname{Spin}\\left(16\\right)} $$<\/div>\u3046\u3061
$$ {\\mathfrak{so}\\left(16\\right)} $$<\/div>\u30ea\u30fc\u4ee3\u6570\u3067\u3059\u3002<\/p>

\u96fb\u8a71\u3057\u305f\u3089

$$ {J_{ij}\\,} $$<\/div>\u30b8\u30a7\u30cd\u30ec\u30fc\u30bf\u30fc\u306e\u30bb\u30c3\u30c8
$$ {\\mathfrak{so}\\left(16\\right)} $$<\/div>\u305d\u3057\u3066
$$ {Q_a\\,} $$<\/div>\u306e 128 \u500b\u306e\u30b3\u30f3\u30dd\u30fc\u30cd\u30f3\u30c8
$$ {S_{16}^+} $$<\/div>\u305d\u3046\u3059\u308c\u3070\u3001\u5b9a\u7fa9\u3059\u308b\u95a2\u4fc2\u3092\u660e\u793a\u7684\u306b\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002
$$ {\\mathfrak{e}_8} $$<\/div>\u3068\u3057\u3066<\/p>
$$ {\\left[J_{ij}, J_{k\\ell}\\right] = \\delta_{jk}J_{i\\ell} – \\delta_{j\\ell}J_{ik} – \\delta_{ik}J_{j\\ell} + \\delta_{i\\ell}J_{jk}\\,} $$<\/div><\/dd><\/dl>

\u540c\u69d8\u306b<\/p>

$$ {\\left[J_{ij}, Q_a\\right] = \\frac14 \\left(\\gamma_i\\gamma_j – \\gamma_j\\gamma_i\\right)_{ab}Q_b\\,} $$<\/div> \u3001<\/dd><\/dl>

\u305d\u308c\u306f\u81ea\u7136\u306a\u4f5c\u7528\u306b\u5bfe\u5fdc\u3057\u307e\u3059

$$ {\\operatorname{so}(16)\\,} $$<\/div>\u30b9\u30d4\u30ce\u30eb\u3067
$$ {S_{16}^+\\,} $$<\/div> \u3002\u6b8b\u308a\u306e\u6574\u6d41\u5b50 (\u5b9f\u969b\u306b\u306f\u6574\u6d41\u5b50\u3067\u3042\u308a\u3001\u53cd\u6574\u6d41\u5b50\u3067\u306f\u3042\u308a\u307e\u305b\u3093) \u306f\u3001\u30b9\u30d4\u30ce\u30eb\u306e\u30b3\u30f3\u30dd\u30fc\u30cd\u30f3\u30c8\u9593\u3067\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002 <\/p>
$$ {\\left[Q_a, Q_b\\right] = \\gamma^{[i}_{ac}\\gamma^{j]}_{cb}J_{ij}\\,} $$<\/div> \u3002<\/dd><\/dl>

\u3053\u308c\u3089\u306e\u5b9a\u7fa9\u304b\u3089\u3001\u30e4\u30b3\u30d3\u6052\u7b49\u5f0f\u304c\u6e80\u305f\u3055\u308c\u3066\u3044\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3067\u304d\u307e\u3059\u3002<\/p>

\u4ee3\u6570<\/h2>

\u30c7\u30a3\u30f3\u30ad\u30f3\u56f3<\/a><\/span><\/span><\/h3>

\u30eb\u30fc\u30c8\u30b7\u30b9\u30c6\u30e0<\/a><\/span><\/span><\/h3>

\u30b7\u30f3\u30d7\u30eb\u306a\u6839\u3067\u5f62\u6210\u3055\u308c\u305f\u30d9\u30fc\u30b9\u306b

$$ {\\mathfrak{so}(16)} $$<\/div> \u3001\u4e00\u65b9\u3067 E \u306e\u6839\u7cfb\u306f\u3001\u306e\u3059\u3079\u3066\u306e\u9806\u5217\u3067\u5f62\u6210\u3055\u308c\u307e\u3059\u3002 <\/p>
$$ {\\left(\\pm 1, \\pm 1, 0, 0, 0, 0, 0, 0\\right)\\,} $$<\/div><\/dd><\/dl>

\u306e\u30eb\u30fc\u30c8\u30b7\u30b9\u30c6\u30e0\u3092\u69cb\u6210\u3059\u308b\u3082\u306e

$$ {\\mathfrak{so}(16)} $$<\/div>\u305d\u3057\u3066\u6301\u3063\u3066\u3044\u307e\u3059
$$ {4\\times\\begin{pmatrix}8\\\\2\\end{pmatrix} = 112\\,} $$<\/div>\u8981\u7d20 ( \u306e\u6b21\u5143\u3067\u3042\u308b 120 \u3092\u53d6\u5f97\u3059\u308b\u306b\u306f\u3001Cartan \u306e 8 \u3064\u306e\u30b8\u30a7\u30cd\u30ec\u30fc\u30bf\u30fc\u3092\u8ffd\u52a0\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059)\u3002<\/p>

\u3055\u3089\u306b\u3001\u3053\u308c\u306b\u810a\u690e\u8868\u73fe\u306e 128 \u306e\u91cd\u307f<\/a><\/span>\u3092\u8ffd\u52a0\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002

$$ {S_{16}^+} $$<\/div>\u306e
$$ {\\mathfrak{so}\\left(16\\right)} $$<\/div> \u3002\u540c\u3058\u57fa\u5e95\u306b\u3042\u308b\u3053\u308c\u3089\u306f\u30d9\u30af\u30c8\u30eb\u3067\u8868\u3055\u308c\u307e\u3059\u3002 <\/p>
$$ {\\left(\\pm\\frac12, \\pm\\frac12, \\pm\\frac12, \\pm\\frac12, \\pm\\frac12, \\pm\\frac12, \\pm\\frac12, \\pm\\frac12\\right)\\,} $$<\/div><\/dd><\/dl>

\u3059\u3079\u3066\u306e\u5ea7\u6a19\u306e\u5408\u8a08\u304c\u5076\u6570\u306b\u306a\u308b<\/a><\/span>\u3088\u3046\u306b\u3057\u307e\u3059\u3002\u756a\u53f7\u304c\u4ed8\u3051\u3089\u308c<\/a><\/span>\u3066\u3044\u307e\u3059

$$ {\\frac12 \\times 2^8 = 128\\,} $$<\/div> \u3002<\/p>

\u3057\u305f\u304c\u3063\u3066\u3001\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002

$$ {112+128=240\\,} $$<\/div>\u30eb\u30fc\u30c8\u3001\u591a\u91cd\u5ea6 1 \u306e\u3059\u3079\u3066\u3002\u8a00\u8449\u306e\u4e71\u7528\u306b\u3088\u308a\u3001\u30bc\u30ed \u30d9\u30af\u30c8\u30eb\u3092\u30ab\u30eb\u30bf\u30f3\u90e8\u5206\u4ee3\u6570\u306b\u95a2\u9023\u4ed8\u3051\u3089\u308c\u305f\u30eb\u30fc\u30c8\u3068\u307f\u306a\u3059\u3053\u3068\u3082\u3042\u308a\u307e\u3059\u3002 E \u304c\u30e9\u30f3\u30af 8 \u3067\u3042\u308b\u305f\u3081\u3001\u30bc\u30ed\u30eb\u30fc\u30c8\u306e\u591a\u91cd\u5ea6\u306f 8 \u306b\u306a\u308a\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u6700\u7d42\u7684\u306b\u3001\u4ee3\u6570\u306e 248 \u500b\u306e\u751f\u6210\u5668\u3092\u5341\u5206\u306b\u8aac\u660e\u3067\u304d\u307e\u3057\u305f\u3002
$$ {\\mathfrak{e}_8} $$<\/div> \u3002<\/p>

\u30ab\u30eb\u30bf\u30f3\u884c\u5217<\/span><\/h3>
$$ { \\begin{pmatrix} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\\\ -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0\\\\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0\\\\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0\\\\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & -1\\\\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0\\\\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0\\\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 2 \\end{pmatrix}} $$<\/div><\/dd><\/dl>

\u7269\u7406\u5b66\u3067\u306f<\/h2>

\u7d20\u7c92\u5b50\u7269\u7406\u5b66<\/a><\/span>\u306b\u304a\u3051\u308b\u5927\u7d71\u4e00<\/a><\/span>\u7406\u8ad6\u306e\u6587\u8108\u3067\u306f\u3001\u30b0\u30eb\u30fc\u30d7 E \u306f\u3001\u4ed6\u306e\u3088\u304f\u8003\u616e\u3055\u308c\u308b\u5927\u7d71\u4e00\u30b0\u30eb\u30fc\u30d7\u3092\u81ea\u7136\u306b\u542b\u3080\u7bc4\u56f2\u3067\u30b2\u30fc\u30b8<\/a><\/span>\u30b0\u30eb\u30fc\u30d7\u306e\u5019\u88dc\u3068\u307f\u306a\u3055\u308c\u308b\u3053\u3068\u304c\u3042\u308a\u307e\u3059\u3002\u9023\u7d9a\u3059\u308b\u30a4\u30f3\u30af\u30eb\u30fc\u30b8\u30e7\u30f3\u306e\u4e0b\u306b\u305d\u308c\u304c\u898b\u3048\u307e\u3059<\/p>

$$ {E_8 \\leftarrow \\operatorname{SO}(10) \\leftarrow \\operatorname{SU}(5) \\leftarrow \\operatorname{SU}(3)\\times\\operatorname{SU}(2)\\times\\operatorname{U}(1)\\,} $$<\/div><\/dd><\/dl>

\u3055\u3089\u306b\u3001E \u30b0\u30eb\u30fc\u30d7\u306f\u5f26\u7406\u8ad6<\/a><\/span>\u3084\u8d85\u91cd\u529b\u306e\u5206\u91ce\u3067\u983b\u7e41\u306b\u767b\u5834\u3057\u307e\u3059\u3002\u30d8\u30c6\u30ed\u30b9\u30c8\u30ea\u30f3\u30b0\u7406\u8ad6<\/a><\/span>\u3067\u306f\u3001\u5b9a\u5f0f\u5316\u306b\u3088\u3063\u3066\u6b21\u306e\u3053\u3068<\/a><\/span>\u304c\u660e\u3089\u304b\u306b\u306a\u308a\u307e\u3059\u3002

$$ {\\textstyle{E_8}\\times\\textstyle{E_8}} $$<\/div> (\u30b3\u30f3\u30d1\u30af\u30c8\u306a\u5f62\u5f0f\u3067) \u30b2\u30fc\u30b8 \u30b0\u30eb\u30fc\u30d7\u3068\u3057\u3066\u3002\u3055\u3089\u306b\u3001\u6700\u5927\u8d85\u91cd\u529b\u304c\u6b21\u5143 8 \u306e\u30c8\u30fc\u30e9\u30b9\u4e0a\u3067\u30b3\u30f3\u30d1\u30af\u30c8\u5316\u3055\u308c\u308b\u3068\u3001\u7d50\u679c\u3068\u3057\u3066\u751f\u3058\u308b\u6b21\u5143 3 \u306e\u7406\u8ad6\u306f\u5927\u57df\u5bfe\u79f0\u6027 E (\u3064\u307e\u308a\u3001\u5c55\u958b\u3055\u308c\u305f\u3001\u307e\u305f\u306f\u6700\u5927\u9650\u975e\u30b3\u30f3\u30d1\u30af\u30c8\u306a\u5f62\u5f0f) \u306b\u306a\u308a\u307e\u3059\u3002\u305d\u306e\u5f8c\u3001\u76ee\u7acb\u305f\u306a\u3044\u30d0\u30fc\u30b8\u30e7\u30f3\u304c\u63d0\u6848\u3055\u308c\u307e\u3057\u305f\u3002
$$ {E_8\\left(\\mathbb{Z}\\right)\\,} $$<\/div>\u3053\u306e\u30b0\u30eb\u30fc\u30d7\u306e\u5bfe\u79f0\u6027\u306f\u3001\u3053\u306e\u6587\u8108\u3067\u306fU-\u53cc\u5bfe\u6027<\/span>\u3068\u547c\u3070\u308c\u308b\u3001 M-\u7406\u8ad6<\/a><\/span>\u306e\u5bfe\u79f0\u6027\u306b\u306a\u308a\u307e\u3059\u3002<\/p>

2007 \u5e74 11 \u6708\u3001\u30a2\u30e1\u30ea\u30ab\u306e\u7269\u7406\u5b66\u8005<\/a><\/span>\u30a2\u30f3\u30c8\u30cb\u30fc \u30ae\u30e3\u30ec\u30c3\u30c8 \u30ea\u30b7\u306f<\/a><\/span>\u3001E \u30b0\u30eb\u30fc\u30d7\u306b\u57fa\u3065\u3044\u305f\u529b\u306e\u7d71\u4e00\u7406\u8ad6\u306b\u95a2\u3059\u308b\u5927\u304d\u306a\u8a71\u984c\u306e\u8ad6\u6587\u3092\u79d1\u5b66\u51fa\u7248\u30b5\u30a4\u30c8ArXiv<\/a><\/span>\u306b\u6295\u7a3f\u3057\u307e\u3057\u305f\u3002<\/p><\/div>

\n\"<\/figure>

\u53c2\u8003\u8cc7\u6599<\/h2>
  1. E8 (\u043c\u0430\u0442\u044d\u043c\u0430\u0442\u044b\u043a\u0430) \u2013 bi\u00e9lorusse<\/a><\/li>
  2. E8 (\u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430) \u2013 tchouvache<\/a><\/li>
  3. E8 (matematik) \u2013 danois<\/a><\/li>
  4. E8 (Gruppe) \u2013 allemand<\/a><\/li>
  5. E8 (mathematics) \u2013 anglais<\/a><\/li>
  6. E8 (matem\u00e1ticas) \u2013 espagnol<\/a><\/li><\/ol><\/div>\n
    \n

    \n E8 (\u6570\u5b66) – \u5b9a\u7fa9\u30fb\u95a2\u9023\u52d5\u753b\n <\/h2>\n
    \n \n
    \n
    \n