The p-centre of Yangians and shifted Yangians

We study the Yangian $Y_n$ associated to the general linear Lie algebra ${\mathfrak{g}\mathfrak{l}}_n$ 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 $U_{\chi }({\mathfrak{g}\mathfrak{l}}_n)$ is Morita equivalent to a quotient of an appropriate choice of shifted Yangian, and so our results will have applications in classical representation theory.