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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 Verma module ： ウィキペディア英語版 Verma module Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics. Verma modules can be used to prove that an irreducible highest weight module with highest weight $\backslash lambda$ is finite-dimensional, if and only if the weight $\backslash lambda$ is dominant and integral. Their homomorphisms correspond to invariant differential operators over flag manifolds. ==Definition of Verma modules== The definition relies on a stack of relatively dense notation. Let $F$ be a field and denote the following: * $\backslash mathfrak$, a semisimple Lie algebra over $F$, with universal enveloping algebra $\backslash mathcal(\backslash mathfrak)$. * $\backslash mathfrak$, a Borel subalgebra of $\backslash mathfrak$, with universal enveloping algebra $\backslash mathcal(\backslash mathfrak)$. * $\backslash mathfrak$, a Cartan subalgebra of $\backslash mathfrak$. We do not consider its universal enveloping algebra. * $\backslash lambda\; \backslash in\; \backslash mathfrak^$ *, a fixed weight. To define the Verma module, we begin by defining some other modules: * $F\_\backslash lambda$, the one-dimensional $F$-vector space (i.e. whose underlying set is $F$ itself) together with a $\backslash mathfrak$-module structure such that $\backslash mathfrak$ acts as multiplication by $\backslash lambda$ and the positive root spaces act trivially. As $F\_\backslash lambda$ is a left $\backslash mathfrak$-module, it is consequently a left $\backslash mathcal(\backslash mathfrak)$-module. * Using the Poincaré–Birkhoff–Witt theorem, there is a natural right $\backslash mathcal(\backslash mathfrak)$-module structure on $\backslash mathcal(\backslash mathfrak)$ by right multiplication of a subalgebra. $\backslash mathcal(\backslash mathfrak)$ is naturally a left $\backslash mathfrak$-module, and together with this structure, it is a $(\backslash mathfrak,\; \backslash mathcal(\backslash mathfrak))$-bimodule. Now we can define the Verma module (with respect to $\backslash lambda$) as : $M\_\backslash lambda\; =\; \backslash mathcal(\backslash mathfrak)\; \backslash otimes\_)\}\; F\_\backslash lambda$ which is naturally a left $\backslash mathfrak$-module (i.e. a representation of $\backslash mathfrak$). The Poincaré–Birkhoff–Witt theorem implies that the underlying vector space of $M\_\backslash lambda$ is isomorphic to : $\backslash mathcal(\backslash mathfrak\_-)\; \backslash otimes\_F\; F\_\backslash lambda$ where $\backslash mathfrak\_-$ is the Lie subalgebra generated by the negative root spaces of $\backslash mathfrak$. 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「Verma module」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.