Twisted Yangians of quasi-split type DI and Drinfeld type current relations

Abstract

We study the twisted Yangian associated with the quasi-split symmetric pair $(\mathfrak{o}_{2n}, \mathfrak{o}_{n-1}\oplus \mathfrak{o}_{n+1})$ of type DI. Working in the transposed $R$-matrix presentation of arXiv:2607.14692, we show that the generating matrix $S(u)$ admits no Gaussian decomposition. We decompose instead the matrix $\widehat{S}(u) = G\,S(u)$, where $G$ is the constant term of $S(u)$. A chain of quasi-determinantal embeddings lowers the rank within this family of pairs and ends at $(\mathfrak{o}_4,\mathfrak{o}_3)$. Together with a direct study of a degenerate block of type AI, this gives a presentation of the special twisted Yangian $SY^{\mathrm{tw}}(\mathfrak{o}_{2n},\mathfrak{o}_{n-1}\oplus \mathfrak{o}_{n+1})$ by Drinfeld type current relations. The two fork nodes of the Dynkin diagram carry Drinfeld's presentation of $Y(\mathfrak{sl}_2)$. All other relations are those of split type DI, except that the Serre relation with a doubled fork node has zero right hand side. For $n\ge3$ we obtain an explicit isomorphism between the $R$-matrix presentation of Guay and Regelskis and the Drinfeld presentation of Lu and Zhang. The Drinfeld generators are expressed through the Gaussian series of $\widehat{S}(u)$. With quasi-split type AIII (Lu and Zhang), this completes the explicit comparison for the classical quasi-split types with a nontrivial diagram involution.