This paper tackles the long-standing problem of quantizing the rational spin Ruijsenaars–Schneider model originating in the work of Krichever and Zabrodin (Russ Math Surv 50:1101, 1995. arXiv:hep-th/9505039). We make use of the technique of quantum Hamiltonian reduction to construct a quantized quiver variety \(\mathfrak {A}_{N,\ell }\) , which is simultaneously the algebra of quantum observables of the rational spin Ruijsenaars–Schneider model of N particles with \(\ell \) spin polarizations. Inside this algebra, we find a loop algebra and Yangian of \(\mathfrak {gl}_\ell \) and conjecture that the algebra \(\mathfrak {A}_{N,\ell }\) can be identified with a truncated Yangian of affine type \(A_{\ell -1}^{(1)}\) . Finally, we use the commutation relations inside \(\mathfrak {A}_{N,\ell }\) to derive a difference equation for eigenstates of the lowest Hamiltonian that reproduces the known quantization of the spinless case when \(\ell =1\) .
