$ı$Hall algebras of weighted projective lines and quantum symmetric pairs

Abstract

The $ı$Hall algebra of a weighted projective line is defined to be the semi-derived Ringel-Hall algebra of the category of $1$-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 $ı$quantum loop algebra, which is a generalization of the $ı$quantum group arising from the quantum symmetric pair of split affine type ADE in its Drinfeld type presentation. The $ı$Hall algebra of the $ı$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 $ı$Hall algebras, explaining the isomorphism of the $ı$quantum group of split affine type A under the two presentations.

V3, 70 pages, split into two papers

References in corpus (4)

• On triangulated orbit categories
• Canonical bases for tensor products and super Kazhdan-Lusztig theory
• Categorification of quantum symmetric pairs I
• Hall algebras and quantum symmetric pairs I: foundations