  In this dissertation we study the coefficients spaces (SAYD modules) of
Hopf-cyclic cohomology theory over a certain family of bicrossed product Hopf
algebras, and we compute the Hopf-cyclic cohomology of such Hopf algebras with
coefficients. We associate a Hopf algebra, what we call a Lie-Hopf algebra, to
any matched pair of Lie groups, Lie algebras and affine algebraic groups via
the semi-dualization procedure of Majid. We then identify the SAYD modules over
Lie-Hopf algebras with the representations and corepresentations of the total
Lie group, Lie algebra or the affine algebraic group of the matched pair. First
we classify the SAYD modules that correspond only to the representations of a
total Lie group (algebra). We call them induced SAYD modules. We then
generalize this identification, focusing on the matched pair of Lie algebras.
We establish a one-to-one correspondence between the SAYD modules over the
Lie-Hopf algebra associated to a matched pair of Lie algebras and certain SAYD
modules over the total Lie algebra. Once the SAYD modules are associated to the
representations and the corepresentations of Lie algebras, nontrivial examples
can be constructed. This way, we illustrate a highly nontrivial 4-dimensional
SAYD module over the Schwarzian Hopf algebra H_{1S}. In addition, we discuss
the periodic cyclic cohomology of Lie-Hopf algebras with nontrivial SAYD
coefficients. We obtain a general van Est isomorphism identifying the periodic
cyclic cohomology of a Lie-Hopf algebra with the (relative) Lie algebra
cohomology of the corresponding total Lie algebra.
