$$\imath $$ Hall algebras of weighted projective lines and quantum symmetric pairs II: injectivity

Abstract

We show that the morphism \(\Omega \) from the \(\imath \)quantum loop algebra \({}^{\text {Dr}}\widetilde{{{\textbf{U}}}}^\imath (L{\mathfrak {g}})\) of split type to the \(\imath \)Hall algebra of the weighted projective line is injective if \({\mathfrak {g}}\) is of finite or affine type. As a byproduct, we use the whole \(\imath \)Hall algebra of the cyclic quiver \(C_n\) to realize the \(\imath \)quantum loop algebra of affine \(\mathfrak {gl}_n\).