Shuffle algebra realizations for modular Yangians

We study the shuffle algebra realization of the positive subalgebra $Y_n^{>}(\mathbb{k})$ of the modular Yangian of classical type over an algebraically closed field $\mathbb{k}$ of characteristic $p>3$. Unlike the characteristic zero case, the natural map from $Y_n^{>}(\mathbb{k})$ to the modular shuffle algebra is not an isomorphism. We determine its kernel and image, which yields a commutative diagram that identifies the restricted Yangian with the small Yangian ${\bar{y}}_n^{>}(\mathbb{k})$ (obtained by reduction modulo $p$ from an integral form of $Y_n^{>}(\mathbb{C})$) via their shuffle algebra realizations. The proofs rely on a modified version of the specialization maps from the characteristic-zero theory that remains valid in positive characteristic.