Reducibility of quasiperiodic cocycles in linear Lie groups

TL;DR

This paper investigates the conditions under which quasiperiodic cocycles in linear Lie groups can be simplified to a reducible form, revealing specific modular reductions for different groups.

Contribution

It establishes that GL(n,C)-reducible cocycles are G-reducible modulo 2 or 1 depending on the Lie group G, providing new insights into reducibility criteria.

Findings

G-valued cocycles reducible in GL(n,C) are G-reducible modulo 2 for certain groups

G-reducibility modulo 1 for unitary groups U(n)

Results clarify the modular conditions for reducibility in various Lie groups

Abstract

Let G be a linear Lie group. We define the G-reducibility of a continuous or discrete cocycle modulo N. We show that a G-valued continuous or discrete cocycle which is GL(n,C)-reducible is in fact G-reducible modulo 2 if G=GL(n,R),SL(n,R),Sp(n,R) or O(n) and modulo 1 if G=U(n).