$\imath$quantum groups of split type via derived Hall algebras

TL;DR

This paper constructs $ extit{i}$quantum groups of split type using derived Hall algebras of 1-periodic complexes, providing a new algebraic realization of these structures.

Contribution

It introduces a novel realization of split type $ extit{i}$quantum groups via derived Hall algebras, linking quantum symmetric pairs to derived categories.

Findings

Realization of $ extit{i}$quantum groups using derived Hall algebras.

Establishment of a connection between quantum symmetric pairs and derived categories.

New algebraic framework for split type $ extit{i}$quantum groups.

Abstract

A quantum symmetric pair consists of a quantum group $\mathbf U$ and its coideal subalgebra ${\mathbf U}^{\imath}_{{\boldsymbol{\varsigma}}}$ (called an $\imath$quantum group) with parameters ${\boldsymbol{\varsigma}}$. In this note, we use the derived Hall algebras of 1-periodic complexes to realize the $\imath$quantum groups ${\mathbf U}^{\imath}_{{\boldsymbol{\varsigma}}}$ of split type.