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In mathematics, a (''B'', ''N'') pair is a structure on groups of Lie type that allows one to give uniform proofs of many results, instead of giving a large number of case-by-case proofs. Roughly speaking, it shows that all such groups are similar to the general linear group over a field. They were invented by the mathematician Jacques Tits, and are also sometimes known as Tits systems. ==Definition== A (''B'', ''N'') pair is a pair of subgroups ''B'' and ''N'' of a group ''G'' such that the following axioms hold: * ''G'' is generated by ''B'' and ''N''. * The intersection, ''H'', of ''B'' and ''N'' is a normal subgroup of ''N''. *The group ''W'' = ''N/H'' is generated by a set ''S'' of elements ''wi'' of order 2, for ''i'' in some non-empty set ''I''. *If ''wi'' is an element of ''S'' and ''w'' is any element of ''W'', then ''wiBw'' is contained in the union of ''BwiwB'' and ''BwB''. *No generator ''wi'' normalizes ''B''. The idea of this definition is that ''B'' is an analogue of the upper triangular matrices of the general linear group ''GL''''n''(''K''), ''H'' is an analogue of the diagonal matrices, and ''N'' is an analogue of the normalizer of ''H''. The subgroup ''B'' is sometimes called the Borel subgroup, ''H'' is sometimes called the Cartan subgroup, and ''W'' is called the Weyl group. The pair (''W'',''S'') is a ''Coxeter system''. The number of generators is called the rank. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「(B, N) pair」の詳細全文を読む スポンサード リンク
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