Twisted Yangians of types BI, CI, DI and Drinfeld type current relations

We study twisted Yangians associated with the split symmetric pairs of types BI, CI and DI. We introduce a new presentation of these algebras, which we call the transposed presentation, governed by a twisted reflection equation that interacts naturally with the Gaussian decomposition of the generating matrix. Working entirely within the $R$-matrix presentation, we derive Drinfeld-type current presentations of the special twisted Yangian $SY^{\mathrm{tw}}(\mathfrak{g}_N)$ and of the extended twisted Yangian $X^{\mathrm{tw}}(\mathfrak{g}_N)$, in which the Serre relations are stated in a closed current form. Extracting coefficients recovers the Drinfeld presentation due to Lu. As a consequence, we establish the isomorphism between the $R$-matrix and Drinfeld presentations of these twisted Yangians conjectured by Lu, Wang and Zhang. As a byproduct, we obtain a tensor product decomposition of $X^{\mathrm{tw}}(\mathfrak{g}_N)$ into the twisted Yangian in the Drinfeld presentation and a polynomial ring in countably many central variables. We also obtain Poincar\'e-Birkhoff-Witt bases in the Drinfeld generators and describe the coideal coproduct on the low Drinfeld modes.