Hopf Cyclic Cohomology, Characteristic Classes of Foliations, and Quantum Invariant of Knots.

Hopf 循环上同调、叶状特征类和结的量子不变量。

• 批准号: 355531-2013
• 负责人: Rangipour, Bahram
• 金额: $ 1.38万
• 依托单位: University of New Brunswick
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635413
• 关键词: Hopf; Cyclic; Cohomology; Characteristic; Classes

Noncommutative Geometry (NCG) is an area of mathematics which applies algebras, including non-commutative ones, in solving geometric problems. Expressions in non-commutative algebras are sensitive to the order of their elements. For example, if you shuffle a deck of cards and then cut it, the result is not the same if you cut it and then shuffle it. Manifolds look locally like points, lines, planes, etc. For example, balloons and tire tubes are manifolds of dimension 2 because you may construct them by stitching together a few plane-like pieces. On the other hand, a non commutative manifold usually comes with two parts which are a manifold and a relation on it that identifies some of its points. It is obvious that any manifold is a non-commutative one but not vice versa. A foliation on a manifold slices it into manifolds with a lower dimension. For example, an onion which is a three dimensional manifold is decomposed into its leaves which are all sphere-like and hence two dimensional. Similarly a cabbage is also decomposed into its leaves. Mathematically, onion and cabbage are the same but their foliations are not equal. By identifying all points of each leaf we obtain a non-commutative manifold. Foliations have applications in many areas of mathematics and physics. Study of foliations is very old and their characterization has been open since 1960s. There have been different theories applied in this area with partial successes. However there are still many fundamental open problems. We plan to solve these open problems via methods of non-commutative geometry. Our primary tools are Hopf algebras and Hopf cyclic cohomology. Hopf algebras are the non-commutative counterpart of symmetries. Hopf cyclic cohomology defines invariants for symmetric non-commutative manifolds. This study not only will benefit the theory of foliations but also enables us to apply foliations and their characteristic classes in many other areas of mathematics such as number theory, quantum field theory, knot theory, and deformations theory. For example, symmetry of a round table is all rotations around the origin in the plane. However the symmetry of a rectangle table is reflections to the coordinate axes in the plane. For example the realizablity conjecture, which speculates that for each class in the Gelfand-Fuks cohomology there exists a distinguished foliation, is still far from being solved. Symmetry in classical geometry is governed by groups, while to study noncommutative spaces one needs Hopf algebras(quantum groups) as a generalization of groups. Hopf cyclic cohomology is a theory which provides invariants of algebras under symmetry of Hopf algebras. The theory was invented by Connes-Moscovic in 1998 and since then has been developed by the proposer and his collaborators. In the first phase of this project we have applied Hopf algebras and their cohomology to develop new theories for characterization of foliations. Some of these theories have no counterpart in the theory of foliations. For instance the twisting via Stable-Anti-Yetter-Drinfeld modules is merely due to Hopf cyclic cohomology. Our recent investigations on this subject make Hopf cyclic theory a new home for characteristic classes of foliations. Our goal is to continue reformulating and solving foliations problems with the help of Hopf cyclic theory. We propose to apply Hopf cyclic cohomology in the study of characteristic classes of foliations from the Noncommutative Geometry point of view. This is another piece of our long term research on the subject by which we plan to reformulate foliations and their invariants. Our ultimate goal is to prove realizablility and recognizability conjectures. Our research so far has established the fact that Hopf algebras are suitable sources of symmetry and infinitesimals for noncommutative spaces. One of the noncommutative spaces on which one can exercise all aspects of Noncommutative Geometry is the space of leaves of a foliation, on which Differential Geometry is powerless. In another parallel but not far project we plan to derive quantum invariant of knots from the category of Yetter-Drinfled modules over quantum groups. We would like to extend our recent results on the enveloping algebras to quantum algebras. Noncommutative geometry developed out of coalescence of operator algebras and differential geometry. This is in contrast to Differential Geometry, where spaces are sets of points and functions on them are secondary objects. This exchange of roles is well understood by the Gelfand-Naimark theorem which establishes the equivalence of commutative C*-algebras and locally compact Hausdorff spaces. However, as a crucial axiom we do not deprive noncommutative algebras of being coordinates algebras of a "spaces". In this project we apply and also develop a wide variety of concepts and objects of Hopf cyclic cohomology including: Hopf algebras of transverse geometries, stable-anti-Yetter-Drinfled modules, cup products, twisted cyclic cocycles, and local index formula.For example consider shuffling and cutting of a deck of cards. Then there is a difference between cutting then shuffling and shuffling and then cutting. Assume that there is a complicated puzzle. Via classical geometry you cannot see the picture. NCG provides you with the notebook of the locations of the pixels. In this example the scattered puzzle together with the notebook resembles an example of noncommutative spaces; while the solved puzzle represents a classical space. To see the meaning of symmetry you may draw a square in a sheet. Then the moves of the sheet with no effect on the square are given by the rotation around the center of the square by 90 degree and the reflection with respect to a line that divides the square into two equal rectangles. The group of symmetries of square comprises the repetition and combination of theses two simple moves.

非交换几何（NCG）是一个应用代数（包括非交换代数）解决几何问题的数学领域。非交换代数中的表达式对元素的阶很敏感。例如，如果你洗一副牌，然后剪切它，如果你剪切它，然后洗它，结果是不一样的。流形局部看起来像点，线，平面等。例如，气球和轮胎管是2维流形，因为你可以通过缝合一些平面状的碎片来构造它们。另一方面，一个非交换流形通常有两个部分，一个是流形，另一个是流形上的一个关系，它可以识别它的一些点。显然，任何流形都是非交换流形，但反之则不然。流形上的叶理把它切割成维数较低的流形。例如，一个洋葱是一个三维流形，它被分解成它的叶子，叶子都是球形的，因此是二维的。同样，卷心菜也会分解成叶子。从数学上讲，洋葱和卷心菜是一样的，但它们的叶子不相等。通过识别每个叶的所有点，我们得到一个非交换流形。叶子在数学和物理学的许多领域都有应用。叶理的研究历史悠久，自20世纪60年代以来，其表征一直是开放的。在这一领域应用了不同的理论，并取得了部分成功。然而，仍然存在许多根本性的开放问题。我们计划通过非交换几何的方法来解决这些开放的问题。我们的主要工具是霍普夫代数和霍普夫循环上同调。Hopf代数是对称的非交换对应。Hopf循环上同调定义了对称非交换流形的不变量。这项研究不仅将有利于理论的叶理，但也使我们能够应用叶理及其特征类在许多其他领域的数学，如数论，量子场论，纽结理论，变形理论。例如，圆桌的对称性是围绕平面中的原点的所有旋转。然而，矩形表的对称性是平面坐标轴的反射。例如，可实现性猜想（realizablity consumption），它推测对于Gelfand-Fuks上同调中的每一个类，都存在一个独特的叶理（foliation），仍然远未解决。经典几何中的对称性是由群控制的，而研究非对易空间则需要Hopf代数（量子群）作为群的推广。Hopf循环上同调是一种在Hopf代数的对称性下给出代数不变量的理论。该理论由Connes-Moscovic于1998年发明，此后由提议者及其合作者发展。在这个项目的第一阶段，我们应用了Hopf代数和他们的上同调发展新的理论表征的叶理。其中有些理论在叶理理论中没有对应的理论。例如，通过Stable-Anti-Yetter-Drinfeld模的扭曲仅仅是由于霍普夫循环上同调。我们最近对这个问题的调查，使霍普夫循环理论的特征类叶理的新家。我们的目标是继续重新制定和解决叶理问题的帮助下，霍普夫循环理论。本文从非对易几何的观点出发，将Hopf循环上同调应用于叶理特征类的研究。这是我们长期研究的另一部分，我们计划通过它来重新制定叶理及其不变量。我们的最终目标是证明可实现性和可识别性。到目前为止，我们的研究已经确立了这样一个事实，即Hopf代数是非交换空间的对称性和无穷小的合适来源。其中一个非交换空间，人们可以行使所有方面的非交换几何是空间的叶子的一个foliation，在这方面微分几何是无能为力的。在另一个并行但不远的项目中，我们计划从量子群上的Yetter-Drinfled模范畴中导出纽结的量子不变量。我们想把我们最近关于包络代数的结果推广到量子代数。非对易几何是由算子代数和微分几何结合而发展起来的。这与微分几何相反，微分几何中空间是点的集合，而它们上的函数是次要对象。这种角色的交换可以通过Gelfand-Naimark定理很好地理解，该定理建立了交换C*-代数和局部紧豪斯多夫空间的等价性。然而，作为一个重要的公理，我们并不剥夺非交换代数的坐标代数的一个“空间”。在这个项目中，我们应用并发展了各种各样的Hopf循环上同调的概念和对象，包括：横截几何的Hopf代数，稳定反Yetter-Drinfled模，杯积，扭循环上循环和局部指数公式。然后有一个不同的切割，然后洗牌和洗牌，然后切割。假设有一个复杂的谜题。通过经典几何学，你看不到这幅图。NCG为您提供像素位置的笔记本。在这个例子中，分散的谜题和笔记本类似于一个非对易空间的例子;而解决的谜题代表一个经典空间。要明白对称的意义，你可以在一张纸上画一个正方形。然后，通过围绕正方形中心旋转90度以及相对于将正方形分成两个相等矩形的线的反射，给出了对正方形没有影响的片材的移动。正方形的对称群就是这两个简单移动的重复和组合。