{"id":2449,"date":"2012-09-05T23:03:18","date_gmt":"2012-09-05T15:03:18","guid":{"rendered":"http:\/\/vanabel.sinaapp.com\/?p=2449"},"modified":"2012-09-05T23:03:18","modified_gmt":"2012-09-05T15:03:18","slug":"%e6%9d%8e%e4%bb%a3%e6%95%b0%e7%9a%84%e4%bc%b4%e9%9a%8f%e8%a1%a8%e7%a4%ba","status":"publish","type":"post","link":"https:\/\/lttt.vanabel.cn\/?p=2449","title":{"rendered":"\u674e\u4ee3\u6570\u7684\u4f34\u968f\u8868\u793a"},"content":{"rendered":"

$
\n\\renewcommand{\\emph}[1]{\\textsf{#1}}
\n\\newcommand{\\Ad}{\\mathrm{Ad}}
\n\\newcommand{\\ad}{\\mathrm{ad}}
\n\\newcommand{\\GL}{\\mathrm{GL}}
\n\\newcommand{\\rd}{\\mathrm{d}}
\n$<\/p>\n

\n
Abstract.<\/span>
\n\u5728\u672c\u6587\u4e2d, \u6211\u5c06\u5bf9~Petersen~\u7684\u9ece\u66fc\u51e0\u4f55\u4e2dp.378\u7684\u5f15\u740669, \u5f15\u740670\u7ed9\u4e00\u70b9\u6ce8\u8bb0. \u5f15\u740669 \u662f\u8bf4\u674e\u4ee3\u6570\u7684\u4f34\u968f\u8868\u793a\u53ef\u4ee5\u7528\u5229\u7528\u674e\u62ec\u53f7\u79ef\u8868\u51fa. \u800c\u5f15\u740670, \u4f5c\u4e3a\u8fd9\u4e2a\u7ed3\u679c\u7684\u63a8\u8bba, \u7ed9\u51fa\u4e86\u4e00\u822c\u7ebf\u6027\u7fa4\u8fd9\u4e2a\u674e\u7fa4\u7684\u674e\u62ec\u53f7\u79ef\u7684\u5177\u4f53\u8868\u8fbe\u5f0f.<\/p>\n
\n

\u5047\u8bbe$G$\u662f\u4e00\u4e2a\u674e\u7fa4, \u5bf9$G$\u4e2d\u4e00\u56fa\u5b9a\u5143$h$, \u5b9a\u4e49$G$\u7684\u4e00\u4e2a\u7531$h$\u51b3\u5b9a\u7684\u5185\u81ea\u540c\u6784<\/strong>$\\alpha_h$ \u5982\u4e0b
\n\\begin{align*}
\n\\alpha_h\\mathpunct{:}G&\\to G\\\\
\nx&\\mapsto hxh^{-1}.
\n\\end{align*}
\n\u5bb9\u6613\u770b\u51fa, $\\alpha_h(x)=R_{h^{-1}}L_h(x)$, \u56e0\u6b64$\\alpha_h=R_{h^{-1}}L_h$.<\/p>\n

\u8bb0$\\alpha_h$\u5728\u5355\u4f4d\u5143$e\\in G$\u5904\u7684\u5207\u6620\u5c04<\/strong>\u4e3a$\\Ad(h)$. \u5373$\\Ad(h)\\mathpunct{:}T_eG\\to T_eG$. \u4e14$\\Ad(h)=D\\alpha_h|_e=DR_{h^{-1}}|_eDL_h|_e$.
\n
\n\u7531\u6b64, \u6211\u4eec\u53c8\u5b9a\u4e49\u4e86\u4e00\u4e2a\u6620\u5c04$\\Ad$,
\n\\begin{align*}
\n\\Ad\\mathpunct{:}G&\\to\\GL(n,T_eG)\\\\
\nh&\\mapsto\\Ad(h),
\n\\end{align*}
\n\u5bb9\u6613\u9a8c\u8bc1, \u5b83\u662f\u674e\u7fa4$G$\u4e0e\u674e\u7fa4$\\GL(n,T_eG)$\u4e4b\u95f4\u7684\u4e00\u4e2a\u540c\u6001, \u79f0\u4e3a\u674e\u7fa4$G$\u7684\u4f34\u968f\u8868\u793a<\/strong>.<\/p>\n

\u518d\u6b21, \u7531$\\Ad$\u7684\u5207\u6620\u5c04, \u53ef\u4ee5\u8bf1\u5bfc\u674e\u7fa4$G$\u4e0e\u674e\u7fa4$\\GL(n,T_eG)$\u7684\u674e\u4ee3\u6570\u95f4\u7684\u540c\u6001. \u5373, \u5b9a\u4e49$\\ad=D\\Ad|_e\\mathpunct{:}T_eG\\to T_I\\GL(n,T_eG)$.<\/p>\n

\u6709\u4e86\u4e0a\u9762\u8fd9\u4e9b\u5b9a\u4e49, \u5f15\u740669\u662f\u8bf4:<\/p>\n

Lemma1.<\/span>(Lemma 69, p378)<\/span>
\n\\[
\n\\ad(u)(v)=[u,v]=[U,V]|_e,
\n\\]
\n\u5176\u4e2d, $u,v\\in T_eG$, \u800c$U,V$\u662f\u5206\u522b\u662f\u7531\u5b83\u4eec\u751f\u6210\u7684\u5de6\u4e0d\u53d8\u5411\u91cf\u573a.
\nProof.<\/span>\u53d6$U$\u7684\u4e00\u4e2a\u6d41$F^t$, \u5373<\/p>\n

\\[\\begin{cases}
\n\\dot F^t(g)|_{t=0}=U_g\\\\
\nF^t(g)|_{t=0}=g.
\n\\end{cases}\\]
\n\u5bb9\u6613\u770b\u51fa,
\n\\begin{align*}
\nU_g&=L_gU_e=L_g\\dot F^t(e)|_{t=0}\\\\
\ng&=L_ge=L_gF^t(e)|_{t=0},
\n\\end{align*}<\/p>\n

\u56e0\u6b64, $L_gF^t(e)$\u548c$F^t(g)$\u6709\u76f8\u540c\u7684\u521d\u503c. \u4ece\u800c $F^t(g)=L_gF^t(e)$, $F^t=R_{F^t(e)}$.
\n\u73b0\u5728,
\n\\begin{align*}
\n\\ad(u)v&=[D\\Ad|_e(u)](v)
\n=\\left[\\left.\\frac{\\rd }{\\rd t}\\right|_{t=0}\\Ad(F^t(e))\\right](v)\\\\
\n&=\\left.\\frac{\\rd }{\\rd t}\\right|_{t=0}\\left[\\Ad(F^t(e))(v)\\right]\\\\
\n&=\\left.\\frac{\\rd}{\\rd t}\\right|_{t=0}\\left[DR_{F^{-t}(e)}DL_{F^t(e)}(v)\\right]\\\\
\n&=\\left.\\frac{\\rd}{\\rd t}\\right|_{t=0}DR_{F^-t(e)}(V|_{F^t(e)})\\\\
\n&=\\left.\\frac{\\rd}{\\rd t}\\right|_{t=0}DF^{-t}(V_{F^t(e)})\\\\
\n&=\\left[L_UV\\right]|_e,
\n\\end{align*}
\n\u6700\u540e\u4e00\u6b65\u662f\u56e0\u4e3a, \u6309\u7167Petersen\u4e66\u4e0a\u7684\u5b9a\u4e49,<\/p>\n

\\[
\nF^{-t}(V|_{F^t(e)})=v+t\\left[L_UV\\right]|_e+o(t).
\n\\]<\/p>\n

\u73b0\u5728, \u6211\u4eec\u5c06\u7528\u4e0a\u9762\u5f15\u7406\u6765\u8ba1\u7b97\u4e00\u822c\u7ebf\u6027\u7fa4\u7684\u674e\u4ee3\u6570.<\/p>\n

\u9996\u5148, \u6211\u4eec\u77e5\u9053, \u5bf9\u4e00\u4e2a\u7ebf\u6027\u7a7a\u95f4$V$, \u5176\u4e0a\u7684\u5168\u4f53\u53ef\u9006\u7ebf\u6027\u53d8\u6362\u6240\u6784\u6210\u7684\u7fa4$\\GL(n,V)$\u662f\u4e00\u4e2a\u674e\u7fa4. \u800c\u4e14\u5b83\u662f$R^{n^2}$\u7684\u4e00\u4e2a\u5f00\u5b50\u6d41\u5f62<\/strong>. \u4e8e\u662f\u5176\u5728\u5355\u4f4d\u5143\u5904\u7684\u5207\u7a7a\u95f4$T_I\\GL(n,V)$\u53ef\u4ee5\u548c\u5176\u81ea\u8eab\u7b49\u540c. \u73b0\u5728, \u5f15\u740670\u662f\u8bf4<\/p>\n

Lemma2.<\/span>(Lemma 70, p378)<\/span>
\n\u5047\u8bbe$x,y\\in T_I\\GL(n,V)$ \u5219:
\n\\[
\n[x,y]=xy-yx,
\n\\]<\/p>\n

\u8fd9\u91cc, \u53f3\u8fb9\u7684$x,y$\u5e94\u770b\u6210\u7ebf\u6027\u53d8\u6362.<\/p><\/div>\n

Proof.<\/span>\u8bbe$x,y$\u751f\u6210\u7684\u5de6\u4e0d\u53d8\u5411\u91cf\u573a\u5206\u522b\u4e3a$X,Y$. \u6ce8\u610f$\\GL(n,V)$\u7684\u5185\u81ea\u540c\u6784$\\alpha_h\\mathpunct{:}x\\mapsto hxh^{-1}$ \u662f\u4e00\u4e2a\u7ebf\u6027\u6620\u5c04, \u4ece\u800c\u5176\u5207\u6620\u5c04, \u5373$\\Ad(h)$, \u6ee1\u8db3$\\Ad(h)(x)=\\alpha_h(x)=hxh^{-1}$.<\/p>\n

\u5176\u6b21, \u7531\u4e8e$\\GL(n,V)$\u662f\u7ebf\u6027\u7a7a\u95f4\u4e2d\u7684\u5f00\u96c6, \u4ece\u800c$X$\u7684\u5c40\u90e8\u6d41\u4e3a$F^t(g)=g(I+tx+o(t))$, \u8fd9\u91cc$x=X|_I$, $g\\in\\GL(n,V)$. \u5229\u7528\u4e0a\u9762\u5f15\u7406\u7684\u7ed3\u679c<\/p>\n

\\begin{align*}
\n[x,y]&=(L_XY)|_I=\\ad(x)(y)=D\\Ad(x)|_I(y)\\\\
\n&=\\left[\\left.\\frac{\\rd}{\\rd t}\\right|_{t=0}\\Ad(F^t(I))\\right](y)\\\\
\n&=\\left.\\frac{\\rd}{\\rd t}\\right|_{t=0}\\left[\\Ad(F^t(I))(y)\\right]\\\\
\n&=\\left.\\frac{\\rd}{\\rd t}\\right|_{t=0}\\left[F^t(I)yF^{-t}(I)\\right]\\\\
\n&=\\left.\\frac{\\rd}{\\rd t}\\right|_{t=0}\\left[
\n(I+tx+o(t))y(I-tx+o(t))
\n\\right]\\\\
\n&=\\left.\\frac{\\rd}{\\rd t}\\right|_{t=0}\\left[
\ny+txy-tyx+o(t)
\n\\right]\\\\
\n&=xy-yx.
\n\\end{align*}\n<\/p><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

$ \\renewcommand{\\emph}[1]{\\textsf{#1}} \\ …<\/p>\n

\u674e\u4ee3\u6570\u7684\u4f34\u968f\u8868\u793a<\/span> \u9605\u8bfb\u66f4\u591a »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12],"tags":[170,800,707,709],"class_list":["post-2449","post","type-post","status-publish","format-standard","hentry","category-mathnotes","tag-170","tag-lecture-notes","tag-707","tag-709"],"_links":{"self":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/2449","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2449"}],"version-history":[{"count":0,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/2449\/revisions"}],"wp:attachment":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2449"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2449"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2449"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}