The cotangent bundle of a matched pair Lie group, and its trivialization, are shown to be a matched pair Lie group. The explicit matched pair decomposition on the trivialized bundle is presented. On the trivialized space, the canonical symplectic two-form and the canonical Poisson bracket are explicitly written. Various symplectic and Poisson reductions are perfomed. The Lie-Poisson bracket is derived. As an example, Lie-Poisson equations on sl(2, C) * are obtained.

In this paper a general van Est type isomorphism is established. The isomorphism is between the Lie algebra cohomology of a bicrossed sum Lie algebra and the Hopf cyclic cohomology of its Hopf algebra. We first prove a one to one correspondence between stable-anti-Yetter-Drinfeld (SAYD) modules over the total Lie algebra and SAYD modules over the associated Hopf algebra. In contrast to the non-general case done in our previous work, here the van Est isomorphism is found at the first level of a natural spectral sequence, rather than at the level of complexes. It is proved that the Connes-Moscovici Hopf algebras do not admit any finite dimensional SAYD modules except the unique one-dimensional one found by Connes-Moscovici in 1998. This is done by extending our techniques to work with the infinite dimensional Lie algebra of formal vector fields. At the end, the one to one correspondence is applied to construct a highly nontrivial four dimensional SAYD module over the Schwarzian Hopf algebra. We then illustrate the whole theory on this example. Finally explicit representative cocycles of the cohomology classes for this example are calculated.

Abstract. Given a matched pair of Lie groups, we show that the tangent bundle of the matched pair group is isomorphic to the matched pair of the tangent groups. We thus obtain the EulerLagrange equations on the trivialized matched pair of tangent groups, as well as the EulerPoincaré equations on the matched pair of Lie algebras. We show explicitly how these equations cover those of the semi-direct product theory. In particular, we study the trivialized, and the reduced Lagrangian dynamics on the group SL(2, C).

To any locally finite representation of a given double crossed sum (product) Lie algebra (group), we associate a stable anti Yetter-Drinfeld (SAYD) module over the bicrossed product Hopf algebra which arises from the semidualization procedure. We prove a van Est isomorphism between the relative Lie algebra cohomology of the total Lie algebra and the Hopf cyclic cohomology of the corresponding Hopf algebra with coefficients in the associated SAYD module.

In the present paper we investigate the noncommutative geometry of a class of algebras, called the Hom-associative algebras, whose associativity is twisted by a homomorphism. We define the Hochschild, cyclic, and periodic cyclic homology and cohomology for this class of algebras generalizing these theories from the associative to the Homassociative setting.