论文标题
大量子组的泊松订单
Poisson orders on large quantum groups
论文作者
论文摘要
我们开发了一个泊松几何框架，用于研究统一根部的所有争议性量子超级群体的表示理论。这是以统一的方式来完成的，通过处理所有杰出的Nichols代数Arxiv的较大类量子双打：1405.6681属于一个参数家族；我们称这些代数\ emph {大}量子组。我们证明，这些量子代数中的每一个都有一个中央的HOPF子代数，从ARXIV的意义上产生了泊松订单：MATH/0201042 ..我们明确描述了基础泊松代数群体和泊松型托有同质空间，该空间是由BoreL borel complect of Complect semimimple eimglaimple Algebraic eimgebraic typept类型的borel均等。所涉及的Poisson代数组和泊松同质空间的几何形状以及其应用于代数$ _ {\ Mathfrak {\ Mathfrak {Q}} \ supset u _ {\ supset u _ {\ Mathfrak {还描述了u _ {\ mathfrak {q}}}^+$。除了所有（多组合）大量子群以外，统一根部的de concini-kac- procesi和大量子超级群体外，我们的框架还包含34维kac-weisfeler的特征性0中的量化，在特征2中，在特征性2和以前的brown contription and Algebras中均置入了三维的算法。泊松支架，在超级情况下是不可能的，因为多达4个发电机有13种额外的Serre关系。我们使用一种依赖于限制和非限制积分形式之间完美配对的新方法。
We develop a Poisson geometric framework for studying the representation theory of all contragredient quantum super groups at roots of unity. This is done in a uniform fashion by treating the larger class of quantum doubles of bozonizations of all distinguished pre-Nichols algebras arXiv:1405.6681 belonging to a one-parameter family; we call these algebras \emph{large} quantum groups. We prove that each of these quantum algebras has a central Hopf subalgebra giving rise to a Poisson order in the sense of arXiv:math/0201042.. We describe explicitly the underlying Poisson algebraic groups and Poisson homogeneous spaces in terms of Borel subgroups of complex semisimple algebraic groups of adjoint type. The geometry of the Poisson algebraic groups and Poisson homogeneous spaces that are involved and its applications to the irreducible representations of the algebras $U_{\mathfrak{q}} \supset U_{\mathfrak{q}}^{\geqslant} \supset U_{\mathfrak{q}}^+$ are also described. Besides all (multiparameter) big quantum groups of De Concini--Kac--Procesi and big quantum super groups at roots of unity, our framework also contains the quantizations in characteristic 0 of the 34-dimensional Kac-Weisfeler Lie algebras in characteristic 2 and the 10-dimensional Brown Lie algebras in characteristic 3. The previous approaches to the above problems relied on reductions to rank two cases and direct calculations of Poisson brackets, which is not possible in the super case since there are 13 kinds of additional Serre relations on up to 4 generators. We use a new approach that relies on perfect pairings between restricted and non-restricted integral forms.
