Isomorphism between twisted $q$-Yangians and affine $\imath$quantum groups: type AI
Kang Lu
TL;DR
The paper establishes an explicit isomorphism between the special twisted $q$-Yangian ${\mathrm{SY}_q^{\rm tw}}(\mathfrak{o}_n)$ and the affine $\imath$quantum group of type AI, using Gauss decomposition of the twist matrix $S(u)$ to define new current generators and compare with the affine $\imath$quantum group in Drinfeld’s current presentation. The main result yields a PBW-type basis for the affine $\imath$quantum group and unifies current- and $R$-matrix realizations for type AI, with low-rank checks in ranks 1 and 2 illustrating the construction. The approach relies on quasi-determinants, Gauss decompositions, and carefully analyzed automorphisms and embeddings, building a bridge between twisted $q$-Yangians and $\imath$quantum groups that has potential applications to open spin chains and boundary integrable models. Collectively, the work consolidates the structural correspondence between Serre-type and Drinfeld-type presentations in this setting and provides concrete tools (PBW bases, Sklyanin determinants) for further representation-theoretic and physical explorations.
Abstract
By employing Gauss decomposition, we establish a direct and explicit isomorphism between the twisted $q$-Yangians (in R-matrix presentation) and affine $\imath$quantum groups (in current presentation) associated to symmetric pair of type AI introduced by Molev-Ragoucy-Sorba and Lu-Wang, respectively. As a corollary, we obtain a PBW type basis for affine $\imath$quantum groups of type AI.