In this paper we study perturbations of constant co-cycles for actions of higher rank semi-simple algebraic groups and their lattices. Roughly speaking, for ergodic actions, Zimmer's co-cycle superrigidity theorems implies that the perturbed cocycle is measurably conjugate to a constant cocycle modulo a compact valued cocycle. The main point of this article is to see that a cocycle which is a c...

In this paper we study perturbations of constant co-cycles for actions of higher rank semi-simple algebraic groups and their lattices. Roughly speaking, for ergodic actions, Zimmer's co-cycle superrigidity theorems implies that the perturbed cocycle is measurably conjugate to a constant cocycle modulo a compact valued cocycle. The main point of this article is to see that a cocycle which is a c...

In this paper we study perturbations of constant co-cycles for actions of higher rank semi-simple algebraic groups and their lattices. Roughly speaking, for ergodic actions, Zimmer's co-cycle superrigidity theorems implies that the perturbed cocycle is measurably conjugate to a constant cocycle modulo a compact valued cocycle. The main point of this article is to see that a cocycle which is a c...

We derive general Novikov-Morse type inequalities in a Conley type framework for flows carrying cocycles, therefore generalizing our results in [FJ2] derived for integral cocycle. The condition of carrying a cocycle expresses the nontriviality of integrals of that cocycle on flow lines. Gradient-like flows are distinguished from general flows carrying a cocycle by boundedness conditions on thes...

We derive general Novikov-Morse type inequalities in a Conley type framework for flows carrying cocycles, therefore generalizing our results in [FJ2] derived for integral cocycle. The condition of carrying a cocycle expresses the nontriviality of integrals of that cocycle on flow lines. Gradient-like flows are distinguished from general flows carrying a cocycle by boundedness conditions on thes...

There is a natural construction which associates to a finitely generated, countable, discrete group G and a 3-cocycle ω of G an inclusion of II1 factors, the so-called diagonal subfactors (with cocycle). In the case when the cocycle is trivial these subfactors are well studied and their standard invariant (or planar algebra) is known. We give a description of the planar algebra of these subfact...

In this paper we study Hopf algebras with a coalgebra projection A ∼= R#ξH and their deformations by an H-bilinear cocycle. If γ is a cocycle for A, then γ can be restricted to a cocycle γR for R, and A γ ∼= RR#ξγH. As examples, we consider liftings of B(V )#K[Γ] where Γ is a finite abelian group, V is a quantum plane and B(V ) is its Nichols algebra, and explicitly construct the cocycle which ...

In a remarkable 1993 paper, R. Sczech defined a group cocycle on GLn(Q) valued in a certain Q-vector space. This cocycle is constructed using conditionally convergent sums over lattices in Euclidean space and is called the Eisenstein cocycle. Sczech showed that certain specializations of the Eisenstein cocycle yield special values of the L-series of totally real fields. We will describe how one...

In this paper we explicitly attach to a geometric classical r-matrix r (not necessarily unitary), a geometric (i.e., set-theoretical) quantum R-matrix R, which is a quantization of r. To accomplish this, we use the language of bijective cocycle 7-tuples, developed by A. Soloviev in the study of set-theoretical quantum R-matrices. Namely, we define a classical version of bijective cocycle 7-tupl...

After a review of some the main results about hyperfinite equivalence relations and their cocycles in the measured setting, we give a definition of a topological AF equivalence relation. We show that every cocycle is cohomologous to a quasi-product cocycle. We then study the problem of determining the quasi-invariant probability measures admitting a given cocycle as their Radon-Nikodym derivative.