  The quiver Yangians were originally defined for the quiver and superpotential
from string theory on general toric Calabi-Yau threefolds, and serve as BPS
algebras of these systems. Their characters reproduce the unrefined BPS
indices, which correspond to classical Donaldson-Thomas (DT) invariants. We
generalize this construction in two directions. First, we show that this
definition extends to arbitrary quivers with potentials. Second, we explain how
to define the characters to incorporate the refined BPS indices, which
correspond to motivic DT invariants. We focus on two main classes of quivers:
the BPS quivers of 4D $N=2$ theories and the quivers from the knot-quiver
correspondence. The entire construction allows for straightforward
generalizations to trigonometric, elliptic, and generalized cohomologies.
