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

Abstract

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 $S{Y}^{\mathrm{tw}}({\mathfrak{g}}_N)$ 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.

44 pages; comments are welcome