  Classical inequalities used in information theory such as those of de Bruijn,
Fisher, and Kullback carry over from the setting of probability theory on
Euclidean space to that of unimodular Lie groups. These are groups that posses
integration measures that are invariant under left and right shifts, which
means that even in noncommutative cases they share many of the useful features
of Euclidean space. In practical engineering terms the rotation group and
Euclidean motion group are the unimodular Lie groups of most interest, and the
development of information theory applicable to these Lie groups opens up the
potential to study problems relating to image reconstruction from irregular or
random projection directions, information gathering in mobile robotics,
satellite attitude control, and bacterial chemotaxis and information
processing. Several definitions are extended from the Euclidean case to that of
Lie groups including the Fisher information matrix, and inequalities analogous
to those in classical information theory are derived and stated in the form of
fifteen small theorems. In all such inequalities, addition of random variables
is replaced with the group product, and the appropriate generalization of
convolution of probability densities is employed.
