$ı$Hall algebras of weighted projective lines and quantum symmetric pairs III: quasi-split type

Abstract

From a category $\mathcal{A}$ with an involution $\varrho$, we introduce $\varrho$-complexes, which are a generalization of (bounded) complexes, periodic complexes and modules of $ı$quiver algebras. The homological properties of the category ${\mathcal{C}}_{\varrho }(\mathcal{A})$ of $\varrho$-complexes are given to make the machinery of semi-derived Ringel-Hall algebras applicable. The $ı$Hall algebra of the weighted projective line $\mathbb{X}$ is the twisted semi-derived Ringel-Hall algebra of ${\mathcal{C}}_{\varrho }(\mathrm{coh}(\mathbb{X}))$, where $\varrho$ is an involution of $\mathrm{coh}(\mathbb{X})$. This $ı$Hall algebra is used to realize the quasi-split $ı$quantum loop algebra, which is a generalization of the $ı$quantum group arising from the quantum symmetric pair of quasi-split affine type ADE in its Drinfeld type presentation.

61 pages