Hopf algebras of transvers geometries and their hopf cyclic cohomology

目前的建议涉及合适的方法和技术的发展，将促进具体计算与一个新的代数机制称为Hopf循环上同。这个装置是由Connes和Moscovici在计算由非交换坐标定义的几何空间的特征不变量的过程中发明的（就像它们出现在量子物理学中一样）。他们的工作揭示了一种特殊类型的对称在处理非交换坐标时的重要性，这种对称由称为Hopf代数的对象编码。我们的建议集中在一类这样的物体上，它们与具有无限多参数的经典对称群密切相关，这些对称群是由Sophus Lie发现并由Elie Cartan在大约100年前研究的)。我们建议开发适当的代数（同调）工具，将允许显式计算他们的Hopf循环上同调。经典群有两类：平的和非平的。在平面情况下，我们将复杂的Hopf代数，称为交叉积Hopf代数，与所讨论的群联系起来。这样一个Hopf代数是由两个较小的Hopf代数组成的。下一步是利用次Hopf代数的同调性，开发计算Hopf代数的Hopf循环上同调的同调工具。证明了Hopf代数的Hopf循环上同调与群的李代数的Gelfand-Fuks上同调是相同的。非平坦的情况是另一种情况。相关的霍普夫代数似乎更复杂；不像扁平的情况下，它不是交叉产品形式。相反，它似乎是由代数和协代数以及它们之间的一些相互作用和兼容性组成的一种新的代数结构。这为涉及Hopf代数、Hopf循环上同调和横向几何的新理论打开了一扇窗。我们提出从Hopf循环上同的观点出发，发展新的理论和工具来研究这些新的Hopf代数。

The present proposal concerns the development of suitable methods and techniques that will facilitate concrete calculations with a new algebraic machinery called Hopf cyclic cohomology. This apparatus was invented by Connes and Moscovici in the process of computing characteristic invariants for geometric spaces defined in terms of non-commuting coordinates (as they appear in quantum physics. Their work brought to light the importance of a specific type of symmetry in dealing with non-commutative coordinates, encoded by objects called Hopf algebras. Our proposal focuses on a class of such objects, which are closely related to classical groups of symmetries with infinitely many parameters, discovered by Sophus Lie and studied by Elie Cartan about 100 years ago). We propose to develop appropriate algebraic (homological) tools that will allow the explicit computation of their Hopf cyclic cohomology. There are two categories of classical groups: flat and non-flat. In the flat case, we associate a sophisticated Hopf algebra, called a bicrossed product Hopf algebra, to the group in question. Such a Hopf algebra is made of two minor Hopf algebras. The next step is to develop homological tools for computing the Hopf cyclic cohomology of the Hopf algebra by using the homology of the minor Hopf algebras. We show that the Hopf cyclic cohomology of the Hopf algebra is the same as the Gelfand-Fuks cohomology of the Lie algebra of the group. The non-flat case is a different situation. The associated Hopf algebra seems to be more sophisticated; unlike the flat case, it is not of bicrossed product form. Instead, it seems to be a new algebraic construction made of an algebra and a coalgebra together with some interactions and compatibilities between them. This opens a window to a new theory involving the theory of Hopf algebras, Hopf cyclic cohomology and transversal geometry. We proposed to develop new theories and tools to study these new Hopf algebras from the Hopf cyclic cohomology point of view.