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

TL;DR

This paper introduces a new algebraic framework called $ ho$-complexes for categories with involution, and uses it to realize quasi-split $ extit{i}$quantum loop algebras via $ extit{i}$Hall algebras of weighted projective lines, extending the theory of quantum symmetric pairs.

Contribution

It develops the concept of $ ho$-complexes and applies semi-derived Ringel-Hall algebra techniques to realize quasi-split $ extit{i}$quantum loop algebras from weighted projective lines.

Findings

Defined $ ho$-complexes generalizing complexes and modules.

Constructed $ extit{i}$Hall algebra of weighted projective lines.

Realized quasi-split $ extit{i}$quantum loop algebra via $ extit{i}$Hall algebra.

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 $\imath$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 $\imath$Hall algebra of the weighted projective line $\mathbb{X}$ is the twisted semi-derived Ringel-Hall algebra of $\mathcal{C}_\varrho({\rm coh}(\mathbb{X}))$, where $\varrho$ is an involution of ${\rm coh}(\mathbb{X})$. This $\imath$Hall algebra is used to realize the quasi-split $\imath$quantum loop algebra, which is a generalization of the $\imath$quantum group arising from the quantum symmetric pair of quasi-split affine type ADE in its Drinfeld type presentation.