Hopf algebras of transvers geometries and their hopf cyclic cohomology

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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.