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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 Yangian ： ウィキペディア英語版 Yangian Yangian is an important structure in modern representation theory, a type of a quantum group with origins in physics. Yangians first appeared in the work of Ludvig Faddeev and his school concerning the quantum inverse scattering method in the late 1970s and early 1980s. Initially they were considered a convenient tool to generate the solutions of the quantum Yang–Baxter equation. The name ''Yangian'' was introduced by Vladimir Drinfeld in 1985 in honor of C.N. Yang. The center of Yangian can be described by quantum determinant. == Description == For any finite-dimensional semisimple Lie algebra ''a'', Drinfeld defined an infinite-dimensional Hopf algebra ''Y''(''a''), called the Yangian of ''a''. This Hopf algebra is a deformation of the universal enveloping algebra ''U''(''a''()) of the Lie algebra of polynomial loops of ''a'' given by explicit generators and relations. The relations can be encoded by identities involving a rational ''R''-matrix. Replacing it with a trigonometric ''R''-matrix, one arrives at affine quantum groups, defined in the same paper of Drinfeld. In the case of the general linear Lie algebra ''gl''''N'', the Yangian admits a simpler description in terms of a single ''ternary'' (or ''RTT'') ''relation'' on the matrix generators due to Faddeev and coauthors. The Yangian Y(''gl''''N'') is defined to be the algebra generated by elements $t\_^$ with 1 ≤ ''i'', ''j'' ≤ ''N'' and ''p'' ≥ 0, subject to the relations :$(t\_^)\; -\; (t\_^)=\; -(t\_^t\_^\; -\; t\_^\; t\_^).$ Defining $t\_^=\backslash delta\_$, setting :$T(z)\; =\; \backslash sum\_\; t\_^\; z^$ and introducing the R-matrix ''R''(''z'') = I + ''z''−1 ''P'' on C''N''$\backslash otimes$C''N'', where ''P'' is the operator permuting the tensor factors, the above relations can be written more simply as the ternary relation: :$\backslash displaystyle(z)T\_(w)\; =\; T\_(w)\; T\_(z)\; R\_(z-w).\}$ The Yangian becomes a Hopf algebra with comultiplication Δ, counit ε and antipode ''s'' given by :$(\backslash Delta\; \backslash otimes\; \backslash mathrm)T(z)=T\_(z)T\_(z),\; \backslash ,\backslash ,\; (\backslash varepsilon\backslash otimes\; \backslash mathrm)T(z)=\; I,\; \backslash ,\backslash ,\; (s\backslash otimes\; \backslash mathrm)T(z)=T(z)^.$ At special values of the spectral parameter $(z-w)$, the ''R''-matrix degenerates to a rank one projection. This can be used to define the quantum determinant of $T(z)$, which generates the center of the Yangian. The twisted Yangian Y−(''gl''''2N''), introduced by G. I. Olshansky, is the co-ideal generated by the coefficients of :$\backslash displaystyle$ where σ is the involution of ''gl''''2N'' given by :$\backslash displaystyleE\_.\}$ Quantum determinant is the center of Yangian. 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「Yangian」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.