. . . "In mathematics, the indefinite orthogonal group, O(p, q) is the Lie group of all linear transformations of an n-dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature (p, q), where n = p + q. It is also called the pseudo-orthogonal group or generalized orthogonal group. The dimension of the group is n(n \u2212 1)/2. The indefinite special orthogonal group, SO(p, q) is the subgroup of O(p, q) consisting of all elements with determinant 1. Unlike in the definite case, SO(p, q) is not connected \u2013 it has 2 components \u2013 and there are two additional finite index subgroups, namely the connected SO+(p, q) and O+(p, q), which has 2 components \u2013 see for definition and discussion. The signature of the form determines the group up to isomorphism; interchanging p with q amounts to replacing the metric by its negative, and so gives the same group. If either p or q equals zero, then the group is isomorphic to the ordinary orthogonal group O(n). We assume in what follows that both p and q are positive. The group O(p, q) is defined for vector spaces over the reals. For complex spaces, all groups O(p, q; C) are isomorphic to the usual orthogonal group O(p + q; C), since the transform changes the signature of a form. This should not be confused with the indefinite unitary group U(p, q) which preserves a sesquilinear form of signature (p, q). In even dimension n = 2p, O(p, p) is known as the ."@en . "Gruppo ortogonale indefinito"@it . . . "\u5E7F\u4E49\u6B63\u4EA4\u7FA4"@zh . . . . . "12446"^^ . "Usually, the word orientation refers to the sign on the volume form, and the sign on that flips or not, depending on even or odd dimensions. This paragraph seems to be talking about two different parity transformations and not orientation. Also, it should be clarified whether these parity transformations are inner automorphisms or not. I think they are, but I'm not sure. Maybe they're only inner in some dimensions and not others?"@en . "\u041D\u0435\u0432\u0438\u0437\u043D\u0430\u0447\u0435\u043D\u0430 \u043E\u0440\u0442\u043E\u0433\u043E\u043D\u0430\u043B\u044C\u043D\u0430 \u0433\u0440\u0443\u043F\u0430"@uk . . . 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"In matematica, il gruppo ortogonale indefinito ovvero gruppo pseudo-ortogonale, denotato con O(p, q) , \u00E8 il gruppo di Lie di tutti gli endomorfismi lineari di uno spazio vettoriale reale n-dimensionale che lasciano invariata una forma bilineare simmetrica di segnatura (p, q), dove n = p + q. La dimensione di questo gruppo \u00E8 n(n \u2212 1)/2."@it . . . . . . . . "Vladimir L. Popov"@en . . "In mathematics, the indefinite orthogonal group, O(p, q) is the Lie group of all linear transformations of an n-dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature (p, q), where n = p + q. It is also called the pseudo-orthogonal group or generalized orthogonal group. The dimension of the group is n(n \u2212 1)/2. In even dimension n = 2p, O(p, p) is known as the ."@en . . . . "In matematica, il gruppo ortogonale indefinito ovvero gruppo pseudo-ortogonale, denotato con O(p, q) , \u00E8 il gruppo di Lie di tutti gli endomorfismi lineari di uno spazio vettoriale reale n-dimensionale che lasciano invariata una forma bilineare simmetrica di segnatura (p, q), dove n = p + q. La dimensione di questo gruppo \u00E8 n(n \u2212 1)/2. Il gruppo ortogonale indefinito speciale, SO(p, q) , \u00E8 il sottogruppo di O(p, q) formato da tutti gli endomorfismi lineari con determinante uguale a 1. Diversamente del caso definito, il gruppo di Lie SO(p, q) non \u00E8 connesso \u2013 infatti ha 2 componenti \u2013 ed inoltre contiene due sottogruppi di indice finito, cio\u00E8 il sottogruppo connesso SO+(p, q) e il sottogruppo a 2 componenti O+(p, q)."@it . . . "1048138055"^^ . . . . . "\u6570\u5B66\u4E0A\uFF0C\u5E7F\u4E49\u6B63\u4EA4\u7FA4\u6216\u79F0\u4F2A\u6B63\u4EA4\u7FA4\u3001\u4E0D\u5B9A\u6B63\u4EA4\u7FA4O(p,q)\u662F\u6240\u6709\u4FDD\u6301n=p+q\u7EF4\u5B9E\u5411\u91CF\u7A7A\u95F4\u4E0A\u7684\u7B26\u53F7\u4E3A (p,q)\u7684\u975E\u9000\u5316\u5BF9\u79F0\u53CC\u7EBF\u6027\u5F62\u5F0F\u7684\u7EBF\u6027\u53D8\u6362\u7EC4\u6210\u7684\u674E\u7FA4\u3002\u8FD9\u4E2A\u7FA4\u7684\u7EF4\u6570\u662Fn(n\u22121)/2\u3002 \u5E7F\u4E49\u7279\u6B8A\u6B63\u4EA4\u7FA4SO(p,q)\u662FO(p,q)\u4E2D\u6240\u6709\u884C\u5217\u5F0F\u4E3A1\u7684\u5143\u7D20\u6784\u6210\u7684\u5B50\u7FA4\u3002 \u5EA6\u91CF\u7684\u7B26\u53F7\uFF08p\u3001q\u5206\u522B\u4E3A\u6B63\u8D1F\u7279\u5F81\u503C\u7684\u4E2A\u6570\uFF09\u5728\u540C\u6784\u7684\u610F\u4E49\u4E0B\u51B3\u5B9A\u8BE5\u7FA4\uFF1B\u4EA4\u6362p\u548Cq\u76F8\u5F53\u4E8E\u5EA6\u91CF\u6539\u53D8\u60EF\u6027\u6307\u6570\uFF0C\u6240\u4EE5\u7ED9\u51FA\u540C\u6837\u7684\u7FA4\u3002\u5982\u679Cp\u6216q\u7B49\u4E8E0\uFF0C\u90A3\u4E48\u540C\u6784\u4E8E\u666E\u901A\u6B63\u4EA4\u7FA4O(n)\u3002\u6211\u4EEC\u5047\u8BBE\u4E0B\u6587\u4E2Dp\u548Cq\u5747\u662F\u6B63\u6574\u6570\u3002 \u7FA4O(p,q)\u5B9A\u4E49\u5728\u5B9E\u5411\u91CF\u7A7A\u95F4\u4E0A\u3002\u5BF9\u4E8E\u8907\u7A7A\u95F4\uFF0C\u6240\u6709\u7FA4O(p,q; C)\u90FD\u540C\u6784\u4E8E\u901A\u5E38\u6B63\u4EA4\u7FA4O(p + q; C)\uFF0C\u56E0\u4E3A\u8907\u5171\u8F6D\u53D8\u6362\u80FD\u6539\u53D8\u4E8C\u6B21\u578B\u7684\u60EF\u6027\u6307\u6570\u3002"@zh . . "V. L."@en . . . . . . . "O/o070300"@en . . "\u041D\u0435\u043E\u043F\u0440\u0435\u0434\u0435\u043B\u0451\u043D\u043D\u0430\u044F \u043E\u0440\u0442\u043E\u0433\u043E\u043D\u0430\u043B\u044C\u043D\u0430\u044F \u0433\u0440\u0443\u043F\u043F\u0430"@ru . . 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\u043E\u0431\u044B\u0447\u043D\u043E\u0439 \u043E\u0440\u0442\u043E\u0433\u043E\u043D\u0430\u043B\u044C\u043D\u043E\u0439 \u0433\u0440\u0443\u043F\u043F\u0435 O(n). \u0414\u0430\u043B\u0435\u0435 \u043C\u044B \u043F\u0440\u0435\u0434\u043F\u043E\u043B\u0430\u0433\u0430\u0435\u043C, \u0447\u0442\u043E p \u0438 q \u043F\u043E\u043B\u043E\u0436\u0438\u0442\u0435\u043B\u044C\u043D\u044B. \u0413\u0440\u0443\u043F\u043F\u0430 \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u044F\u0435\u0442\u0441\u044F \u0434\u043B\u044F \u0432\u0435\u043A\u0442\u043E\u0440\u043D\u044B\u0445 \u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432 \u043D\u0430\u0434 \u0432\u0435\u0449\u0435\u0441\u0442\u0432\u0435\u043D\u043D\u044B\u043C\u0438 \u0447\u0438\u0441\u043B\u0430\u043C\u0438. \u0414\u043B\u044F \u043A\u043E\u043C\u043F\u043B\u0435\u043A\u0441\u043D\u044B\u0445 \u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432 \u0432\u0441\u0435 \u0433\u0440\u0443\u043F\u043F\u044B 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"Indefinite orthogonal group"@en . . . . . . . . "293609"^^ . . . . . . . . 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"December 2020"@en . . . . . . "Orthogonal group"@en . "Popov"@en . . . . "\u6570\u5B66\u4E0A\uFF0C\u5E7F\u4E49\u6B63\u4EA4\u7FA4\u6216\u79F0\u4F2A\u6B63\u4EA4\u7FA4\u3001\u4E0D\u5B9A\u6B63\u4EA4\u7FA4O(p,q)\u662F\u6240\u6709\u4FDD\u6301n=p+q\u7EF4\u5B9E\u5411\u91CF\u7A7A\u95F4\u4E0A\u7684\u7B26\u53F7\u4E3A (p,q)\u7684\u975E\u9000\u5316\u5BF9\u79F0\u53CC\u7EBF\u6027\u5F62\u5F0F\u7684\u7EBF\u6027\u53D8\u6362\u7EC4\u6210\u7684\u674E\u7FA4\u3002\u8FD9\u4E2A\u7FA4\u7684\u7EF4\u6570\u662Fn(n\u22121)/2\u3002 \u5E7F\u4E49\u7279\u6B8A\u6B63\u4EA4\u7FA4SO(p,q)\u662FO(p,q)\u4E2D\u6240\u6709\u884C\u5217\u5F0F\u4E3A1\u7684\u5143\u7D20\u6784\u6210\u7684\u5B50\u7FA4\u3002 \u5EA6\u91CF\u7684\u7B26\u53F7\uFF08p\u3001q\u5206\u522B\u4E3A\u6B63\u8D1F\u7279\u5F81\u503C\u7684\u4E2A\u6570\uFF09\u5728\u540C\u6784\u7684\u610F\u4E49\u4E0B\u51B3\u5B9A\u8BE5\u7FA4\uFF1B\u4EA4\u6362p\u548Cq\u76F8\u5F53\u4E8E\u5EA6\u91CF\u6539\u53D8\u60EF\u6027\u6307\u6570\uFF0C\u6240\u4EE5\u7ED9\u51FA\u540C\u6837\u7684\u7FA4\u3002\u5982\u679Cp\u6216q\u7B49\u4E8E0\uFF0C\u90A3\u4E48\u540C\u6784\u4E8E\u666E\u901A\u6B63\u4EA4\u7FA4O(n)\u3002\u6211\u4EEC\u5047\u8BBE\u4E0B\u6587\u4E2Dp\u548Cq\u5747\u662F\u6B63\u6574\u6570\u3002 \u7FA4O(p,q)\u5B9A\u4E49\u5728\u5B9E\u5411\u91CF\u7A7A\u95F4\u4E0A\u3002\u5BF9\u4E8E\u8907\u7A7A\u95F4\uFF0C\u6240\u6709\u7FA4O(p,q; C)\u90FD\u540C\u6784\u4E8E\u901A\u5E38\u6B63\u4EA4\u7FA4O(p + q; C)\uFF0C\u56E0\u4E3A\u8907\u5171\u8F6D\u53D8\u6362\u80FD\u6539\u53D8\u4E8C\u6B21\u578B\u7684\u60EF\u6027\u6307\u6570\u3002"@zh . . . . . . .