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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 canonical basis ： ウィキペディア英語版 canonical basis In mathematics, a canonical basis is a basis of an algebraic structure that is canonical in a sense that depends on the precise context: * In a coordinate space, and more generally in a free module, it refers to the standard basis defined by the Kronecker delta. * In a polynomial ring, it refers to its standard basis given by the monomials, $(X^i)\_i$. * For finite extension fields, it means the polynomial basis. * In linear algebra, it refers to a set of ''n'' linearly independent generalized eigenvectors of an ''n''×''n'' matrix $A$, if the set is composed entirely of Jordan chains. ==Representation theory== In representation theory there are several basis that are called "canonical", e.g. Lusztig's canonical basis and closely related Kashiwara's crystal basis in quantum groups and their representations. There is a general concept underlying these basis: Consider the ring of integral Laurent polynomials $\backslash mathcal:=\backslash mathbb()$ with its two subrings * A dualization operation, i.e. a bijection $F\backslash to\; F$ of order two that is $\backslash overline$-semilinear and will be denoted by $\backslash overline$ as well. If a precanonical structure is given, then one can define the $\backslash mathcal^$ submodule $F^\; :=\; \backslash sum\; \backslash mathcal^\; t\_j$ of $F$. A ''canonical basis at $v=0$'' of the precanonical structure is then a $\backslash mathcal$-basis $(c\_i)\_$ of $F$ that satisfies: * $\backslash overline=c\_i$ and * $c\_i\; \backslash in\; \backslash sum\_\; \backslash mathcal^+\; t\_j$and $c\_i\; \backslash equiv\; t\_i\; \backslash mod\; vF^+$ for all $i\backslash in\; I$. A ''canonical basis at $v=\backslash infty$'' is analogously defined to be a basis $(\backslash widetilde\_i)\_$ that satisfies * $\backslash overline=\backslash widetilde\_i$ and * $\backslash widetilde\_i\; \backslash in\; \backslash sum\_\; \backslash mathcal^-\; t\_j$ and $\backslash widetilde\_i\; \backslash equiv\; t\_i\; \backslash mod\; v^F^-$ for all $i\backslash in\; I$. The naming "at $v=\backslash infty$" alludes to the fact $\backslash lim\_\; v^\; =0$ and hence the "specialization" $v\backslash mapsto\backslash infty$ corresponds to quotienting out the relation $v^=0$. One can show that there exists at most one canonical basis at v=0 (and at most one at $v=\backslash infty$) for each precanonical structure. A sufficient condition for existence is that the polynomials $r\_\backslash in\backslash mathcal$ defined by $\backslash overline=\backslash sum\_i\; r\_\; t\_i$ satisfy $r\_=1$ and $r\_\backslash neq\; 0\; \backslash implies\; i\backslash leq\; j$. A canonical basis at v=0 ($v=\backslash infty$) induces an isomorphism from $\backslash textstyle\; F^+\backslash cap\; \backslash overline\; =\; \backslash sum\_i\; \backslash mathbbc\_i$ to $F^+/vF^+$ ($\backslash textstyle\; F^\; \backslash cap\; \backslash overline\backslash widetilde\_i\; \backslash to\; F^/v^\; F^$ respectively). 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「canonical basis」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.