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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 List of finite-dimensional Nichols algebras ： ウィキペディア英語版 List of finite-dimensional Nichols algebras In mathematics, a Nichols algebra is a Hopf algebra in a braided category assigned to a object ''V'' in this category (e.g. a braided vector space). The Nichols algebra is a quotient of the tensor algebra of ''V'' enjoying a certain universal property and is typically infinite-dimensional. Nichols algebras appear naturally in any pointed Hopf algebra and enabled their classification in important cases.〔Andruskiewitsch, Schneider: ''Pointed Hopf algebras'', New directions in Hopf algebras, 1–68, Math. Sci. Res. Inst. Publ., 43, Cambridge Univ. Press, Cambridge, 2002.〕 The most well known examples for Nichols algebras are the Borel parts $U\_q(\backslash mathfrak)^+$ of the infinite-dimensional quantum groups when ''q'' is no root of unity, and the first examples of finite-dimensional Nichols algebras are the Borel parts $u\_q(\backslash mathfrak)^+$ of the Frobenius–Lusztig kernel (''small quantum group'') when ''q'' is a root of unity. The following article lists all known finite-dimensional Nichols algebras $\backslash mathfrak(V)$ where $V$ is a Yetter–Drinfel'd module over a finite group $G$, where the group is generated by the support of $V$. For more details on Nichols algebras see Nichols algebra. * There are two major cases: * * $G$ abelian, which implies $V$ is diagonaly braided $x\_i\backslash otimes\; x\_j\backslash mapsto\; q\_x\_j\backslash otimes\; x\_i$. * * $G$ nonabelian. * The rank is the number of irreducible summands $V=\backslash bigoplus\_\; V\_i$ in the semisimple Yetter–Drinfel'd module $V$. * The irreducible summands $V\_i=\backslash mathcal\_^\backslash chi$ are each associated to a conjugacy class $()\backslash subset\; G$ and an irreducible representation $\backslash chi$ of the centralizer $\backslash operatorname(g)$. * To any Nichols algbera there is by 〔 attached * * a generalized root system and a Weyl groupoid. These are classified in.〔 * * In particular several Dynkin diagrams (for inequivalent types of Weyl chambers). Each Dynkin diagram has one vertex per irreducible $V\_i$ and edges depending on their braided commutators in the Nichols algebra. * The Hilbert series of the graded algebra $\backslash mathfrak(V)$ is given. An observation is that it factorizes in each case into polynomials $(n)\_t:=1+t+t^2+\backslash cdots\; +\; t^$. We only give the Hilbert series and dimension of the Nichols algebra in characteristic $0$. Note that a Nichols algebra only depends on the braided vector space $V$ and can therefore be realized over many different groups. Sometimes there are two or three Nichols algebras with different $V$ and non-isomorphic Nichols algebra, which are closely related (e.g. cocycle twists of each other). These are given by different conjugacy classes in the same column. == State of classification == (as of 2015) 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「List of finite-dimensional Nichols algebras」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.