翻訳と辞書 Words near each other ・ Regulation of chemicals ・ Regulation of electronic cigarettes ・ Regulation of emotion ・ Regulation of Financial Services (Land Transactions) Act 2005 ・ Regulation of food and dietary supplements by the U.S. Food and Drug Administration ・ Regulation of gastric function ・ Regulation of gene expression ・ Regular clergy ・ Regular conditional probability ・ Regular constraint ・ Regular Democratic Organization ・ Regular Division of the Plane ・ Regular dodecahedron ・ Regular economy ・ Regular element ・ Regular element of a Lie algebra ・ Regular embedding ・ Regular expression ・ Regular extension ・ Regular Force ・ Regular Fries ・ Regular grammar ・ Regular Grand Lodge of Macedonia ・ Regular graph ・ Regular grid ・ Regular Guys ・ Regular Hadamard matrix ・ Regular haircut ・ Regular homotopy ・ Regular icosahedron Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Regular element of a Lie algebra ： ウィキペディア英語版 Regular element of a Lie algebra In mathematics, a regular element of a Lie algebra or Lie group is an element whose centralizer has dimension as small as possible. In the specific case of ''n''x''n'' matrices over an algebraically closed field (such as the complex numbers), an element ''M'' is regular if and only if its Jordan normal form contains a single Jordan block for each eigenvalue. In that case, the centralizer is the set of polynomials of degree less than ''n'' evaluated at the matrix ''M'', and therefore the centralizer has dimension ''n'' (but it is not necessarily an algebraic torus). If the matrix ''M'' is diagonalisable, then it is regular if and only if there are ''n'' different eigenvalues. To see this, notice that ''M'' will commute with any matrix ''P'' that stabilises each of its eigenspaces. If there are ''n'' different eigenvalues, then this happens only if ''P'' is diagonalisable on a same basis as ''M'' is, and in fact ''P'' is a linear combination of the first ''n'' powers of ''M'' in this case, so that the centralizer is an algebraic torus of complex dimension ''n'' (and of dimension ''2n'' as a real manifold); since this is the smallest possible dimension of a centralizer in this case, such a matrix ''M'' is regular. However if there are equal eigenvalues, then the centralizer, which is the product of the general linear groups of the eigenspaces of ''M'', has strictly larger dimension, and ''M'' is not regular in this case. For a connected compact Lie group ''G'', and its Lie algebra ''g'', the regular elements can also be described explicitly. In ''g'' they form an open and dense subset. In ''G'', the regular elements form an open dense subset also; and if ''T'' is a maximal torus of ''G'', the elements ''t'' of ''T'' that are regular in ''G'' determine the regular elements of ''G'', which make up the union of the conjugacy classes in ''G'' of regular elements in ''T''. The regular elements ''t'' are themselves explicitly given as the complement of a set in ''T'', determined by the adjoint action of ''G'', and making up a union of subtori.〔(Mark R. Sepanski, ''Compact Lie groups'' (2007), p. 156. )〕 ==References== * 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「Regular element of a Lie algebra」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.