Drinfeld presentation of affine iquantum groups via iHopf algebras

Abstract

Affine quantum groups $\widetilde{\mathbf{U}}$ and affine iquantum groups $\widetilde{\mathbf {U}}^\imath$ admit two complementary descriptions: a Serre-type presentation and a Drinfeld (current) presentation. While iquantum groups $\widetilde{\mathbf {U}}^\imath$ are typically defined via the Serre presentation, their Drinfeld presentations are substantially more subtle and less well understood. In this paper, we give a conceptual derivation of the Drinfeld presentation of $\widetilde{\mathbf {U}}^\imath$ starting directly from the Drinfeld presentation of $\widetilde{\mathbf {U}}$, working within the framework of iHopf algebras. This approach leads naturally to the definition of iquantum affinization associated with arbitrary Kac-Moody algebras $\mathfrak{g}$. We further show that, in this general setting, these iquantum affinizations can be realized as coideal subalgebras of quantum affinizations with respect to the Drinfeld coproduct. Finally, in the affine case, we describe the imaginary root vectors $Θ_{i,r}$ of $\widetilde{\mathbf {U}}^\imath$ explicitly in terms of the root vectors $ψ^\pm_{i,r}$ of $\widetilde{\mathbf {U}}$, thereby extending the results of Li and Przeździecki for certain affine types of $\widetilde{\mathbf {U}}^\imath$.