Cyclic homology and quantum group symmetry

代数几何和整体分析表明，几何空间(流形、簇等)上的部分几何和拓扑可以有效地用适当的函数环来表示。非对易几何的目的是更进一步，并将所得到的理论纯代数地推广到一般非对易环上。一方面，它展示了某些概念和结果可以公式化的内在背景和一般性，另一方面，它通过将它们应用于特定的非对易环，导致了从数论到理论物理的各种主题的联系。同源技术在这一理论中发挥着核心作用，尤其是在阿兰·康尼斯的方法中。他的计划围绕着对Atiyah-Singer指数定理的深远概括，以及对经典陈一鸣性格的复杂类比。在概念方面，Connes最有影响力的发现可能是循环同调，它是非对易几何框架下De Rham理论的巧妙替代。提出的研究项目的背景是尝试将这些方法应用于通过形变量子化得到的代数。后者形式化了从一个机械系统到量子力学中的对应系统的通道，并将某些非对易代数附加到流形或仿射簇上的泊松结构上。将它应用于李群和代数群产生了量子群，在过去的25年里，量子群已经发现了几个应用，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已经确立的机制进行实质性的概括。充分非平凡的泊松结构产生了一个模类，它在量子水平上由所考虑的代数的某种自同构来表示(更多细节见支持的情况)。很明显，这种自同构可以在几个地方并入到理论中，这是很自然的，有几个原因，但整体情况仍然不清楚。拟议的项目将研究将模性并入非对易几何的某些方面，特别是循环同调本身。这似乎是非对易几何发展的自然下一步。另一方面，广义方法在量子群中的应用可以为Woronowicz的协变微积分理论或具有量子群对称性的物理模型的构建和研究提供新的刺激。

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.