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Cyclic homology and quantum group symmetry

循环同调性和量子群对称性

基本信息

批准号：
EP/E043267/1
负责人：
Ulrich Kraehmer
金额：
$ 31.56万
依托单位：
University of Glasgow
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2007
资助国家：
英国
起止时间：
2007 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FE043267%2F1
关键词：
Cyclic homology quantum group symmetry

项目摘要

Algebraic geometry and global analysis demonstrate that parts of geometry and topology can be reformulated effectively in terms of suitable rings of functions on geometric spaces (manifolds, varieties etc.). Noncommutative geometry aims to go further and to extend the resulting theory purely algebraically towards general noncommutative rings. On one hand this displays the intrinsic setting and generality in which certain concepts and results can be formulated, and on the other hand it led through their application to specific noncommutative rings to connections to subjects ranging from number theory to theoretical physics. Homological techniques play a central role in this theory, and especially in the approach of Alain Connes. His programme is centred around far-reaching generalisations of the Atiyah-Singer index theorem and the involved analogue of the classical Chern character. On the conceptual side, Connes' most influential discovery was probably cyclic homology, a subtle substitute of de Rham theory in the framework of noncommutative geometry.The background of the proposed research project is the attempt to apply these methods to algebras obtained by deformation quantisation. The latter formalises the passage from a mechanical system to its counterpart in quantum mechanics and attaches certain noncommutative algebras to Poisson structures on manifolds or affine varieties. Applying this to Lie groups and algebraic groups yields quantum groups that have found in the last 25 years several applications especially in knot theory and in quantum statistical mechanics. As recent work of several authors indicates, this attempt could lead to substantial generalisations of Connes' well-established machinery. Sufficiently nontrivial Poisson structures give rise to a modular class which is represented on the quantum level by a certain automorphism of the algebra under consideration (see the Case of Support for more details). It became clear that this automorphism can be incorporated at several places into the theory and that this is natural for several reasons, but the overall picture is still unclear.The proposed project will investigate some aspects of this incorporation of modularity into noncommutative geometry, focusing in particular on cyclic homology itself. This seems a natural next step in the development of noncommutative geometry. On the other hand, the applications of the generalised methods to quantum groups could provide new stimulations for example for Woronowicz's theory of covariant differential calculi or for the construction and study of physical models with quantum group symmetry.

代数几何和全局分析表明，部分几何和拓扑可以有效地重新表述为几何空间（流形，簇等）上的函数环。非对易几何的目标是更进一步，并将由此产生的理论纯粹代数地扩展到一般的非对易环。一方面，这显示了某些概念和结果可以公式化的内在设置和一般性，另一方面，它通过它们对特定非交换环的应用，与从数论到理论物理的学科建立了联系。同调技术在这一理论中起着核心作用，特别是在阿兰·康纳斯的方法中。他的计划是围绕着深远的概括的阿蒂亚-辛格指数定理和参与模拟的经典陈省身字符。在概念方面，康纳斯的最有影响力的发现可能是循环同源性，一个微妙的替代德拉姆理论的框架noncommutative geometrics.The背景下提出的研究项目是试图将这些方法应用到代数获得的变形quantisation。后者形式化了从力学系统到量子力学中对应系统的通道，并将某些非交换代数与流形或仿射簇上的泊松结构联系起来。将此应用于李群和代数群产生了量子群，在过去的25年中，量子群已经发现了一些应用，特别是在纽结理论和量子统计力学中。正如几位作者最近的工作所表明的那样，这种尝试可能会导致对康纳斯完善的机制的实质性概括。足够非平凡的泊松结构产生一个模类，它在量子水平上由所考虑的代数的某种自同构表示（更多细节见支撑的情况）。很明显，这种自同构可以在几个地方纳入理论，这是自然的，有几个原因，但总体情况仍然不清楚。拟议的项目将调查这种纳入模块性到非交换几何的某些方面，特别是集中在循环同源本身。这似乎是发展非对易几何的自然的下一步。另一方面，量子群的广义方法的应用可以提供新的刺激，例如Woronowicz的理论的协变微分演算或建设和研究的物理模型与量子群对称性。