Shuffle algebra realizations for modular Yangians

Abstract

We study the shuffle algebra realization of positive modular Yangians of classical type over an algebraically closed field of characteristic $p>3$. We show that, unlike in characteristic zero, the natural shuffle homomorphism has a nontrivial kernel. Its image is characterized by a $p$-wheel condition, while its kernel is precisely the ideal generated by the $p$-th powers of the Lyndon root vectors. This identifies the corresponding quotient with the small Yangian arising from a $\mathbb{Z}[\frac{1}{2}]$-integral form. As part of the construction, we establish a PBW basis for the integral form and obtain the PBW theorem and $p$-center results for the Drinfeld presentation of modular Yangians.

arXiv commentv1: 18 pages, comments are welcome! v2: 24 pages, we extend the results to all classical types. v3: 30pages, we add the PBW theorem and $p$-center results for the Drinfeld presentation of modular Yangians