r$_\infty$-Matrices, triangular L$_\infty$-bialgebras, and quantum$_\infty$ groups

TL;DR

This paper introduces a homotopy analogue of triangular Lie bialgebras aiming to extend quantum group theory into the realm of homotopy algebras, leading to the concept of quantum$_$-groups.

Contribution

It proposes a new homotopical framework for quantum groups, generalizing classical notions to homotopy algebra contexts.

Findings

Defines homotopy triangular Lie bialgebras

Develops a homotopical version of quantum groups

Lays groundwork for future research in homotopy quantum algebra

Abstract

A homotopy analogue of the notion of a triangular Lie bialgebra is proposed with a goal of extending the basic notions of theory of quantum groups to the context of homotopy algebras and, in particular, introducing a homotopical generalization of the notion of a quantum group, or quantum$_\infty$-group.