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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク E8 lattice ： ウィキペディア英語版 E8 lattice In mathematics, the E8 lattice is a special lattice in R8. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8. The name derives from the fact that it is the root lattice of the E8 root system. The norm〔In this article, the ''norm'' of a vector refers to its length squared (the square of the ordinary norm).〕 of the E8 lattice (divided by 2) is a positive definite even unimodular quadratic form in 8 variables, and conversely such a quadratic form can be used to construct a positive-definite, even, unimodular lattice of rank 8. The existence of such a form was first shown by H. J. S. Smith in 1867, and the first explicit construction of this quadratic form was given by A. Korkine and G. Zolotareff in 1873. The E8 lattice is also called the Gosset lattice after Thorold Gosset who was one of the first to study the geometry of the lattice itself around 1900. ==Lattice points== The E8 lattice is a discrete subgroup of R8 of full rank (i.e. it spans all of R8). It can be given explicitly by the set of points Γ8 ⊂ R8 such that *all the coordinates are integers or all the coordinates are half-integers (a mixture of integers and half-integers is not allowed), and *the sum of the eight coordinates is an even integer. In symbols, :$\backslash Gamma\_8\; =\; \backslash left\backslash )^8\; :\; x\_i\; \backslash equiv\; 0\backslash ;(\backslash mbox2)\backslash right\backslash \}.$ It is not hard to check that the sum of two lattice points is another lattice point, so that Γ8 is indeed a subgroup. An alternative description of the E8 lattice which is sometimes convenient is the set of all points in Γ′8 ⊂ R8 such that *all the coordinates are integers and the sum of the coordinates is even, or *all the coordinates are half-integers and the sum of the coordinates is odd. In symbols, :$\backslash Gamma\_8\text{'}\; =\; \backslash left\backslash )^8\; :\; x\_i\}\; \backslash equiv\; 2x\_1\; \backslash equiv\; 2x\_2\; \backslash equiv\; 2x\_3\; \backslash equiv\; 2x\_4\; \backslash equiv\; 2x\_5\; \backslash equiv\; 2x\_6\; \backslash equiv\; 2x\_7\; \backslash equiv\; 2x\_8\backslash ;(\backslash mbox2)\backslash right\backslash \}.$ :$\backslash Gamma\_8\text{'}\; =\; \backslash left\backslash \; \backslash equiv\; 0(\backslash mbox2)\backslash right\backslash \}$ \cup \left\)^8 : x_i} \equiv 1(\mbox2)\right\}. The lattices Γ8 and Γ′8 are isomorphic and one may pass from one to the other by changing the signs of any odd number of coordinates. The lattice Γ8 is sometimes called the ''even coordinate system'' for E8 while the lattice Γ8' is called the ''odd coordinate system''. Unless we specify otherwise we shall work in the even coordinate system. 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「E8 lattice」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.