Cyclic homology and quantum group symmetry

代数几何和全局分析表明，几何空间（流形、变异等）上的部分几何和拓扑可以用合适的函数环有效地重新表述。非交换几何的目标是更进一步，将结果理论在纯代数上扩展到一般的非交换环。一方面，这显示了某些概念和结果可以公式化的内在背景和普遍性，另一方面，它通过将它们应用于特定的非交换环，以及从数论到理论物理等学科的联系。同源技术在这一理论中起着核心作用，特别是在阿兰·科恩斯的方法中。他的计划集中在对Atiyah-Singer指数定理的深远推广和对经典陈氏特征的模拟。在概念方面，Connes最有影响力的发现可能是循环同调，这是在非交换几何框架中对de Rham理论的微妙替代。提出的研究项目的背景是尝试将这些方法应用于变形量化获得的代数。后者形式化了从力学系统到量子力学对应系统的传递，并将某些非交换代数附加到流形或仿射变体上的泊松结构上。将此应用于李群和代数群产生量子群，这些量子群在过去25年中已经发现了一些应用，特别是在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Connes已经建立的机制进行实质性的推广。充分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（参见支持的情况了解更多细节）。很明显，这种自同构可以在理论的几个地方被纳入，这是很自然的，有几个原因，但总体情况仍然不清楚。提议的项目将研究将模块化结合到非交换几何中的一些方面，特别关注循环同调本身。这似乎是非交换几何发展的自然下一步。另一方面，将广义方法应用于量子群可以为Woronowicz的协变微分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.