Shuffle algebra realizations for restricted Yangians

TL;DR

This paper explores the shuffle algebra realization of the positive subalgebra of the Yangian for rak{sl}_n over fields of characteristic p, revealing differences from the characteristic zero case and characterizing the kernel and image of the homomorphism.

Contribution

It determines the kernel and image of the shuffle algebra homomorphism for the restricted Yangian in characteristic p, extending the understanding of Yangian realizations in modular settings.

Findings

Kernel is generated by the p-center of the Yangian.

Image consists of elements satisfying a wheel condition related to characteristic p.

Provides a shuffle algebra realization for the restricted Yangian.

Abstract

We study the shuffle algebra realization of the positive subalgebra $Y_n^{>}(\mathbb{k})$ of the Yangian associated to $\mathfrak{sl}_n$ over an algebraically closed field $\mathbb{k}$ of characteristic $p>2$. In contrast to the characteristic zero case, the natural homomorphism from $Y_n^{>}(\mathbb{k})$ to the modular shuffle algebra $W^{(n)}(\mathbb{k})$ is not an isomorphism. We determine its kernel and image, showing that the kernel is precisely the ideal generated by the $p$-center of $Y_n^{>}(\mathbb{k})$, while the image consists of elements satisfying an additional wheel condition related to the characteristic $p$, thus providing a shuffle algebra realization for the restricted Yangian $Y_n^{>,[p]}$. The proof relies on the specialization maps approach and the construction of the small Yangian $\bar{y}^{>}_n(\mathbb{k})$, obtained by the reduction modulo $p$ method from an integral form $\mathbf{Y}_n^>$ of the Yangian $Y_n^{>}$ associated to $\mathfrak{sl}_n$ over $\mathbb{C}$.