Hopf algebras of transvers geometries and their hopf cyclic cohomology

目前的建议涉及发展合适的方法和技术，这将有利于具体的计算与一个新的代数机器称为霍普夫循环上同调。这个装置是由Connes和Moscovici在计算几何空间的特征不变量的过程中发明的，几何空间是根据非交换坐标定义的（因为它们出现在量子物理学中）。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，这种坐标由称为Hopf代数的对象编码。我们的建议集中在一类这样的对象，这是密切相关的经典群的对称性与无限多的参数，发现了索菲斯李和研究埃利嘉当约100年前）。我们建议开发适当的代数（同调）工具，将允许显式计算其Hopf循环上同调。经典群有两类：平坦群和非平坦群。在平坦的情况下，我们将一个复杂的Hopf代数，称为bicrossed产品的Hopf代数，组的问题。这样的一个Hopf代数是由两个次Hopf代数组成的。下一步是开发同调工具，通过使用次Hopf代数的同调来计算Hopf代数的Hopf循环上同调。证明了该群的Hopf代数的Hopf循环上同调与该群的李代数的Gelfand-Fuks上同调是相同的。非平坦的情况是不同的情况。相关的霍普夫代数似乎更复杂，不像平面的情况下，它不是双交叉产品的形式。相反，它似乎是一个新的代数结构，由一个代数和一个余代数以及它们之间的一些相互作用和相容性组成。这打开了一个窗口，一个新的理论，涉及理论的霍普夫代数，霍普夫循环上同调和横向几何。我们建议发展新的理论和工具来研究这些新的Hopf代数的Hopf循环上同调的角度来看。

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.