𝚤Hall algebras of weighted projective lines and quantum symmetric pairs

The ı<!-- ı --> Hall algebra of a weighted projective line is defined to be the semi-derived Ringel-Hall algebra of the category of 1 1 -periodic complexes of coherent sheaves on the weighted projective line over a finite field. We show that this Hall algebra provides a realization of the ı<!-- ı --> quantum loop algebra, which is a generalization of the ı<!-- ı --> quantum group arising from the quantum symmetric pair of split affine type ADE in its Drinfeld type presentation. The ı<!-- ı --> Hall algebra of the ı<!-- ı --> quiver algebra of split affine type A was known earlier to realize the same algebra in its Serre presentation. We then establish a derived equivalence which induces an isomorphism of these two ı<!-- ı --> Hall algebras, explaining the isomorphism of the ı<!-- ı --> quantum group of split affine type A under the two presentations.