Braid group action and quasi-split affine iquantum groups III

TL;DR

This paper completes the Drinfeld presentation of quasi-split affine iquantum groups of type AIII, constructing root vectors and establishing relations that define their algebraic structure.

Contribution

It provides the final piece in understanding the Drinfeld presentation for type AIII^{( au)}_{2r} quasi-split affine iquantum groups, including root vector construction and relations.

Findings

Constructed basic real and imaginary v-root vectors.

Established relations among root vectors.

Provided a Drinfeld presentation for the algebra.

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 ${\rm AIII}^{(\tau)}_{2r}$. 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 $\widetilde{{\mathbf U}}^\imath$.