  A new type of quantum transfer matrix, arising as a Cholesky factor for the
steady state density matrix of a dissipative Markovian process associated with
the boundary-driven Lindblad equation for the isotropic spin-1/2 Heisenberg
(XXX) chain, is presented. The transfer matrix forms a commuting family of
non-Hermitian operators depending on the spectral parameter which is
essentially the strength of dissipative coupling at the boundaries. The
intertwining of the corresponding Lax and monodromy matrices is performed by an
infinitely dimensional Yang-Baxter R-matrix which we construct explicitly and
which is essentially different from the standard XXX R-matrix. We also discuss
a possibility to construct Bethe Ansatz for the spectrum and eigenstates of the
non-equilibrium steady state density operator. Furthermore, we indicate the
existence of a deformed R-matrix in the infinitely-dimensional auxiliary space
for the anisotropic XXZ spin-1/2 chain which in general provides a sequence of
new, possibly quasi-local, conserved quantities of the bulk XXZ dynamics.
