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

TL;DR

The paper constructs a parabolic (block Gauss) presentation for extended Yangians of types $B$ and $C$, parameterized by symmetric compositions $\nu$ of $N$, and proves its equivalence with the standard BC RTT and Drinfeld presentations. Central to the method is a block RTT framework and a block Gauss decomposition, which yield new parabolic generators $\mathcal{D}_a(u)$, $\mathcal{E}_{ab}(u)$, and $\mathcal{F}_{ba}(u)$ and an embedding theorem $\Psi_p$ that reduces the analysis to small block sizes. The authors derive explicit parabolic relations for both odd and even symmetric compositions, including exchange, commutation, and Serre-type relations, and they express the center via a product of traces of Gauss blocks, providing a concrete center formula in terms of parabolic data. These results unify and extend the understanding of BC-type Yangians, connect to finite $W$-algebras in type A via parabolic subalgebras, and pave the way for analogous treatments in type $D$; the presented framework also yields explicit center expressions useful for representation-theoretic explorations.

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 $ν$ 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 $ν$-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.

Theorems & Definitions (128)

• Theorem 2.1
• proof
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• Lemma 2.5
• proof
• Proposition 2.6
• proof