Quiver algebras and their representations for arbitrary quivers

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 are related 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 explore how one can refine the characters of the quiver Yangian to incorporate the refined BPS indices, which are related to motivic DT invariants. We focus on two main classes of quivers: the BPS quivers of 4D $\mathcal{N}$ \( \mathcal{N} \) = 2 theories and the quivers from the knot-quiver correspondence. The entire construction allows for straightforward generalizations to trigonometric, elliptic, and generalized cohomologies.