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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=572600
• 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）是应用代数（包括非交换代数）解决几何问题的一个数学领域。非交换代数中的表达式对其元素的顺序很敏感。例如，如果你洗牌，然后切牌，结果是不一样的，如果你切牌，然后洗牌。流形在局部看起来像点、线、平面等。例如，气球和轮胎管是2维的流形，因为你可以通过将一些平面状的碎片拼接在一起来构造它们。另一方面，一个非交换流形通常由两个部分组成，一个流形和一个关系，这个关系确定了流形上的一些点。很明显，任何流形都是不可交换的，反之则不然。流形上的叶化将流形切成低维的流形。例如，一个三维流形的洋葱被分解成它的叶子，这些叶子都是球形的，因此是二维的。同样，卷心菜也被分解成叶子。从数学上讲，洋葱和卷心菜是一样的，但它们的叶子不相等。通过识别每个叶上的所有点，我们得到了一个非交换流形。叶理在数学和物理的许多领域都有应用。对叶理的研究非常古老，它们的特征自20世纪60年代以来一直开放。在这一领域应用了不同的理论，并取得了部分成功。然而，仍有许多根本性的问题有待解决。我们计划用非交换几何的方法来解决这些开放问题。我们的主要工具是Hopf代数和Hopf循环上同。Hopf代数是对称的非交换对应物。Hopf循环上同调定义了对称非交换流形的不变量。