The $p$-centre of Yangians and shifted Yangians

We study the Yangian $Y_n$ associated to the general linear Lie algebra $\mathfrak{gl}_n$ over a field of positive characteristic, as well as its shifted analog $Y_n(\sigma)$. Our main result gives a description of the centre of $Y_n(\sigma)$: it is a polynomial algebra generated by its Harish-Chandra centre (which lifts the centre in characteristic zero) together with a large $p$-centre. Moreover, $Y_n(\sigma)$ is free as a module over its center. In future work, it will be seen that every reduced enveloping algebra $U_\chi(\mathfrak{gl}_n)$ is Morita equivalent to a quotient of an appropriate choice of shifted Yangian, and so our results will have applications in classical representation theory.