  By using the algebraic construction outlined in \cite{CGRS}, we introduce
several Markov processes related to the ${\mathcal{U}}_q(\mathfrak{su}(1,1))$
quantum Lie algebra. These processes serve as asymmetric transport models and
their algebraic structure easily allows to deduce duality properties of the
systems. The results include: (a) the asymmetric version of the Inclusion
Process, which is self-dual; (b) the diffusion limit of this process, which is
a natural asymmetric analogue of the Brownian Energy Process and which turns
out to have the symmetric Inclusion Process as a dual process; (c) the
asymmetric analogue of the KMP Process, which also turns out to have a
symmetric dual process. We give applications of the various duality relations
by computing exponential moments of the current.
