AI 中文总结
研究BI、CI、DI型分裂对称对相关的扭曲杨代数，通过新的转置表示，在R - 矩阵表示内推导其Drinfeld型电流表示，建立了与Drinfeld表示的同构，还得到张量积分解、Poincaré - Birkhoff - Witt基及余理想余积。
AI 中文摘要
我们研究与BI、CI和DI型分裂对称对相关的扭曲杨代数。引入一种新的表示，即转置表示，由扭曲反射方程控制，它与生成矩阵的高斯分解自然相互作用。在R - 矩阵表示内，推导特殊扭曲杨代数$SY^{\mathrm{tw}}(\mathfrak{g}_N)$和扩展扭曲杨代数$X^{\mathrm{tw}}(\mathfrak{g}_N)$的Drinfeld型电流表示，其中Serre关系以封闭电流形式给出。提取系数可恢复Lu的Drinfeld表示。建立了Lu、Wang和Zhang猜想的这些扭曲杨代数的R - 矩阵和Drinfeld表示之间的同构。还得到了$X^{\mathrm{tw}}(\mathfrak{g}_N)$的张量积分解，获得了Drinfeld生成元的Poincaré - Birkhoff - Witt基并描述了低Drinfeld模上的余理想余积。
英文摘要
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.
Comments44 pages; comments are welcome
