$\imath$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 $^{\texttt{Dr}}\widetilde{\mathbf{U}}(L\mathfrak{g})$ 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 realise the $\imath$quantum loop algebra of affine $\mathfrak{gl}_n$.