Minimalistic Presentation and Coideal Structure of Twisted Yangians
Kang Lu
TL;DR
The paper constructs a Drinfeld-type, minimalistic presentation for the twisted Yangian ${}^h extstyle{ Y}$ associated with split symmetric pairs and embeds it as a right coideal subalgebra of the Yangian ${ Y}$ via an explicit injective map $ $. It proves that ${}^h extstyle{ Y}$ is isomorphic to the twisted Yangian ${}^h extstyle{ Y}_{ j}$ in $J$-presentation, and provides a precise correspondence between Drinfeld generators and their coproducts in ${ Y}$, including an associated graded description aligning with $U( rak g[z]^{reveoldsymbol om})$. A minimalistic presentation for ${}^h extstyle{ Y}$ is established, with generators $h_{i,1}$, $b_{i,0}$, $b_{i,1}$ and relations capturing the core Serre structure; special low-rank types require an extra relation. The work also derives coproduct estimates for the Drinfeld generators in terms of the Yangian generators, enabling restriction to ${}^h extstyle{ Y}$-modules and the development of boundary $q$-character-like data, with extensions to quasi-split and $q$-deformed settings suggested as future directions.
Abstract
We introduce a minimalistic presentation for the twisted Yangian ${}^\imath\mathscr Y$ associated with split symmetric pairs (or Satake diagrams) introduced in arXiv:2406.05067 via a Drinfeld type presentation. As applications, we establish an injective algebra homomorphism from ${}^\imath\mathscr Y$ to the Yangian $\mathscr Y$, thereby identifying ${}^\imath\mathscr Y$ as a right coideal subalgebra of $\mathscr Y$ and proving its isomorphism with the twisted Yangian in the $J$ presentation. Furthermore, we provide estimates for the Drinfeld generators of ${}^\imath\mathscr Y$ and describe their coproduct images in terms of the Drinfeld generators of $\mathscr Y$ under this identification.