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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 Generalized Verma module ： ウィキペディア英語版 Generalized Verma module In mathematics, generalized Verma modules are a generalization of a (true) Verma module,〔Named after Daya-Nand Verma.〕 and are objects in the representation theory of Lie algebras. They were studied originally by James Lepowsky in the nineteen seventies. The motivation for their study is that their homomorphisms correspond to invariant differential operators over generalized flag manifolds. The study of these operators is an important part of the theory of parabolic geometries. ==Definition== Let $\backslash mathfrak$ be a semisimple Lie algebra and $\backslash mathfrak$ a parabolic subalgebra of $\backslash mathfrak$. For any irreducible finite-dimensional representation $V$ of $\backslash mathfrak$ we define the generalized Verma module to be the relative tensor product :$M\_(\backslash mathfrak)\backslash otimes\_)\}\; V$. The action of $\backslash mathfrak$ is left multiplication in $\backslash mathcal(\backslash mathfrak)$. If λ is the highest weight of V, we sometimes denote the Verma module by $M\_\}(\backslash lambda)$ makes sense only for $\backslash mathfrak$-dominant and $\backslash mathfrak$-integral weights (see weight) $\backslash lambda$. It is well known that a parabolic subalgebra $\backslash mathfrak$ of $\backslash mathfrak$ determines a unique grading $\backslash mathfrak=\backslash oplus\_^k\; \backslash mathfrak\_j$ so that $\backslash mathfrak=\backslash oplus\_\; \backslash mathfrak\_j$. Let $\backslash mathfrak\_-:=\backslash oplus\_\; \backslash mathfrak\_j$. It follows from the Poincaré–Birkhoff–Witt theorem that, as a vector space (and even as a $\backslash mathfrak\_-$-module and as a $\backslash mathfrak\_0$-module), :$M\_(\backslash mathfrak\_-)\backslash otimes\; V$. In further text, we will denote a generalized Verma module simply by GVM. 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「Generalized 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.