Braid group action and quasi-split affine iquantum groups III
Ming Lu, Xiaolong Pan, Weiqiang Wang, Weinan Zhang
TL;DR
This work completes the Drinfeld-type presentation for the quasi-split affine iquantum group $\widetilde{\mathbf U}^\imath$ of type ${\rm AIII}_{2r}^{(\tau)}$ by constructing explicit real and imaginary $v$-root vectors and establishing their relations across rank-one embeddings arising from the three possible affine rank-one substructures. It develops a unified Drinfeld framework, both in commutator form and generating-function form, and proves that the newly introduced Drinfeld data surjects onto and injects into $\widetilde{\mathbf U}^\imath$, using ibraid group actions, translation invariance, and rank-one isomorphisms. The main contributions include the identification of affine rank-one subalgebras compatible with ibraid symmetries, the definition of higher-rank $v$-root vectors, and a complete verification of the Drinfeld-type relations, thus completing the Drinfeld presentation for all quasi-split affine iquantum groups. The results pave the way for geometric realizations, connections to quantum symmetric pairs, and potential applications to quantum integrable systems with boundary conditions and related Coulomb-branch structures.
Abstract
This is the last of three papers on Drinfeld presentations of quasi-split affine iquantum groups $\widetilde{\mathbf U}^\imath$, settling the remaining type ${\rm AIII}^{(τ)}_{2r}$. This type distinguishes itself among all quasi-split affine types in having 3 relative root lengths. Various basic real and imaginary $v$-root vectors for $\widetilde{\mathbf U}^\imath$ are constructed, giving rise to affine rank one subalgebras of $\widetilde{\mathbf U}^\imath$ associated with simple roots in the finite relative root system. We establish the relations among these $v$-root vectors and show that they provide a Drinfeld presentation of $\widetilde{\mathbf U}^\imath$.