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 Poincaré-Birkhoff-Witt bases in the Drinfeld generators and describe the coideal coproduct on the low Drinfeld modes.