  The pentagram map was introduced by R. Schwartz in 1992 and is now one of the
most renowned discrete integrable systems. In the present paper we prove that
this map, as well as all its known integrable multidimensional generalizations,
can be seen as refactorization-type mappings in the Poisson-Lie group of
pseudo-difference operators. This brings the pentagram map into the rich
framework of Poisson-Lie groups, both describing new structures and simplifying
and revealing the origin of its known properties. In particular, for
multidimensional pentagram maps the Poisson-Lie group setting provides new Lax
forms with a spectral parameter and, more importantly, invariant Poisson
structures in all dimensions, the existence of which has been an open problem
since the introduction of those maps. Furthermore, for the classical pentagram
map our approach naturally yields its combinatorial description in terms of
weighted directed networks and cluster algebras.
