Parabolic presentations of Yangian in types $B$ and $C$

TL;DR

This paper introduces a new parabolic presentation of the extended Yangian for types B and C, using block matrix techniques to derive generators, relations, and a formula for the center, enhancing understanding of its algebraic structure.

Contribution

It provides the first parabolic presentation of the extended Yangian in types B and C, including a systematic derivation of generators, relations, and a new formula for the center.

Findings

Derived generators and relations via Gauss decomposition.

Established a block matrix RTT presentation.

Obtained a new formula for the center of the Yangian.

Abstract

We establish a parabolic presentation of the extended Yangian $\X(\mathfrak{g}_{N})$ associated with the Lie algebras $\mathfrak{g}_{N}$ of type $B$ and $C$, parameterized by a symmetric composition $\nu$ of $N$. By formulating a block matrix version of the RTT presentation of $\X(\mathfrak{g}_{N})$, we systematically derive the generators and relations through the Gauss decomposition of the generator matrix in $\nu$-block form. Furthermore, leveraging this parabolic presentation, we obtain a novel formula for the center of $\X(\mathfrak{g}_{N})$, offering new insights into its structure.