  A family of quantum cluster algebras is introduced and studied. In general,
these algebras are new, but subclasses have been studied previously by other
authors. The algebras are indexed by double partitions or double flag
varieties. Equivalently, they are indexed by broken lines $L$. By grouping
together neighboring mutations into quantum line mutations we can mutate from
the cluster algebra of one broken line to another. Compatible pairs can be
written down. The algebras are equal to their upper cluster algebras. The
variables of the quantum seeds are given by elements of the dual canonical
basis.
  This is the final version, where some arguments have been expanded and/or
improved and several typos corrected. Full bibliographic details: Journal of
Algebra (2012), pp. 172-203 DOI information: 10.1016/j.jalgebra.2012.09.015
