  We present a systematic approach for constructing steady state density
operators of Markovian dissipative evolution for open quantum chain models with
integrable bulk interaction and boundary incoherent driving. The construction
is based on fundamental solutions of the quantum Yang-Baxter equation
pertaining to quantum algebra symmetries and their quantizations
(q-deformations). In particular, we facilitate a matrix-product state
description, by resorting to generic spin-s infinite-dimensional solutions
associated with non-compact spins, serving as ancillary degrees of freedom.
After formally deriving already known solutions for the anisotropic spin-1/2
Heisenberg chain from first symmetry principles, we obtain a class of solutions
belonging to interacting quantum gases with SU(N)-symmetric Hamiltonians, using
a restricted set of incoherent boundary jump processes, and point out how new
non-trivial generalizations emerge from twists of quantum group structures.
Finally, we discuss possibilities of analytic calculation of observables by
employing algebraic properties of associated auxiliary vertex operators.
