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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 Crystal bases ： ウィキペディア英語版 Crystal base In algebra, a crystal base or canonical base is a base of a representation, such that generators of a quantum group or semisimple Lie algebra have a particularly simple action on it. Crystal bases were introduced by and (under the name of canonical bases). ==Definition== As a consequence of the defining relations for the quantum group $U\_q(G)$, $U\_q(G)$ can be regarded as a Hopf algebra over $(q)$, the field of all rational functions of an indeterminate ''q'' over $\backslash Bbb\; Q$. For simple root $\backslash alpha\_i$ and non-negative integer $n$, define $e\_i^\; =\; e\_i^n/()\_!$ and $f\_i^\; =\; f\_i^n/()\_!$ (specifically, $e\_i^\; =\; f\_i^\; =\; 1$). In an integrable module $M$, and for weight $\backslash lambda$, a vector $u\; \backslash in\; M\_$ (''i.e.'' a vector $u$ in $M$ with weight $\backslash lambda$) can be uniquely decomposed into the sums * $u\; =\; \backslash sum\_^\backslash infty\; f\_i^\; u\_n\; =\; \backslash sum\_^\backslash infty\; e\_i^\; v\_n,$ where $u\_n\; \backslash in\; \backslash mathrm(e\_i)\; \backslash cap\; M\_$, $v\_n\; \backslash in\; \backslash mathrm(f\_i)\; \backslash cap\; M\_$, $u\_n\; \backslash ne\; 0$ only if $n\; +\; \backslash frac\; \backslash ge\; 0$, and $v\_n\; \backslash ne\; 0$ only if $n\; -\; \backslash frac\; \backslash ge\; 0$. Linear mappings $\backslash tilde\_i\; :\; M\; \backslash to\; M$ and $\backslash tilde\_i\; :\; M\; \backslash to\; M$ can be defined on $M\_$ by * $\backslash tilde\_i\; u\; =\; \backslash sum\_^\backslash infty\; f\_i^\; u\_n\; =\; \backslash sum\_^\backslash infty\; e\_i^\; v\_n,$ * $\backslash tilde\_i\; u\; =\; \backslash sum\_^\backslash infty\; f\_i^\; u\_n\; =\; \backslash sum\_^\backslash infty\; e\_i^.\; v\_n$ Let $A$ be the integral domain of all rational functions in $(q)$ which are regular at $q\; =\; 0$ (''i.e.'' a rational function $f(q)$ is an element of $A$ if and only if there exist polynomials $g(q)$ and $h(q)$ in the polynomial ring $()$ such that $h(0)\; \backslash ne\; 0$, and $f(q)\; =\; g(q)/h(q)$). A crystal base for $M$ is an ordered pair $(L,B)$, such that *$L$ is a free $A$-submodule of $M$ such that $M\; =\; (q)\; \backslash otimes\_A\; L;$ *$B$ is a $\backslash Bbb\; Q$-basis of the vector space $L/qL$ over $\backslash Bbb\; Q,$ *$L\; =\; \backslash oplus\_\; L\_$ and $B\; =\; \backslash sqcup\_\; B\_$, where $L\_\; =\; L\; \backslash cap\; M\_$ and $B\_\; =\; B\; \backslash cap\; (L\_/qL\_),$ *$\backslash tilde\_i\; L\; \backslash subset\; L$ and $\backslash tilde\_i\; L\; \backslash subset\; L\; \backslash text\; i\; ,$ *$\backslash tilde\_i\; B\; \backslash subset\; B\; \backslash cup\; \backslash$ and $\backslash tilde\_i\; B\; \backslash subset\; B\; \backslash cup\; \backslash \backslash text\; i,$ *$\backslash textb\; \backslash in\; B\backslash textb\text{'}\; \backslash in\; B,\backslash texti,\backslash quad\backslash tilde\_i\; b\; =\; b\text{'}\backslash text\backslash tilde\_i\; b\text{'}\; =\; b.$ To put this into a more informal setting, the actions of $e\_i\; f\_i$ and $f\_i\; e\_i$ are generally singular at $q\; =\; 0$ on an integrable module $M$. The linear mappings $\backslash tilde\_i$ and $\backslash tilde\_i$ on the module are introduced so that the actions of $\backslash tilde\_i\; \backslash tilde\_i$ and $\backslash tilde\_i\; \backslash tilde\_i$ are regular at $q\; =\; 0$ on the module. There exists a $(q)$-basis of weight vectors $\backslash tilde$ for $M$, with respect to which the actions of $\backslash tilde\_i$ and $\backslash tilde\_i$ are regular at $q\; =\; 0$ for all ''i''. The module is then restricted to the free $A$-module generated by the basis, and the basis vectors, the $A$-submodule and the actions of $\backslash tilde\_i$ and $\backslash tilde\_i$ are evaluated at $q\; =\; 0$. Furthermore, the basis can be chosen such that at $q\; =\; 0$, for all $i$, $\backslash tilde\_i$ and $\backslash tilde\_i$ are represented by mutual transposes, and map basis vectors to basis vectors or 0. A crystal base can be represented by a directed graph with labelled edges. Each vertex of the graph represents an element of the $\backslash Bbb\; Q$-basis $B$ of $L/qL$, and a directed edge, labelled by ''i'', and directed from vertex $v\_1$ to vertex $v\_2$, represents that $b\_2\; =\; \backslash tilde\_i\; b\_1$ (and, equivalently, that $b\_1\; =\; \backslash tilde\_i\; b\_2$), where $b\_1$ is the basis element represented by $v\_1$, and $b\_2$ is the basis element represented by $v\_2$. The graph completely determines the actions of $\backslash tilde\_i$ and $\backslash tilde\_i$ at $q\; =\; 0$. If an integrable module has a crystal base, then the module is irreducible if and only if the graph representing the crystal base is connected (a graph is called "connected" if the set of vertices cannot be partitioned into the union of nontrivial disjoint subsets $V\_1$ and $V\_2$ such that there are no edges joining any vertex in $V\_1$ to any vertex in $V\_2$). For any integrable module with a crystal base, the weight spectrum for the crystal base is the same as the weight spectrum for the module, and therefore the weight spectrum for the crystal base is the same as the weight spectrum for the corresponding module of the appropriate Kac–Moody algebra. The multiplicities of the weights in the crystal base are also the same as their multiplicities in the corresponding module of the appropriate Kac–Moody algebra. It is a theorem of Kashiwara that every integrable highest weight module has a crystal base. Similarly, every integrable lowest weight module has a crystal base. 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「Crystal base」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.