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We study a new process, which we call ASEP(q, j), where particles move asymmetrically on a one-dimensional integer lattice with a bias determined by (Formula presented.) and where at most (Formula presented.) particles per site are allowed. The process is constructed from a (Formula presented.)-dimensional representation of a quantum Hamiltonian with (Formula presented.) invariance by applying a suitable ground-state transformation. After showing basic properties of the process ASEP(q, j), we prove self-duality with several self-duality functions constructed from the symmetries of the quantum Hamiltonian. By making use of the self-duality property we compute the first q-exponential moment of the current for step initial conditions (both a shock or a rarefaction fan) as well as when the process is started from a homogeneous product measure

A generalized asymmetric exclusion process with $$U_q(\mathfrak {sl}_2)$$ stochastic duality