Braided cofree Hopf algebras and quantum multi-brace algebras

TL;DR

This paper systematically constructs Hopf algebra structures on braided cofree coalgebras, introduces quantum multi-brace algebras as a unifying framework, and applies these concepts to quantize important algebraic structures like quantum quasi-shuffle algebras.

Contribution

It introduces quantum multi-brace algebras, generalizing braided and B-infinity algebras, providing a new framework for quantization of algebraic structures.

Findings

Construction of Hopf algebra structures on braided cofree coalgebras

Introduction of quantum multi-brace algebras as a unifying framework

Application to quantum quasi-shuffle algebras

Abstract

We give a systematic construction of Hopf algebra structures on braided cofree coalgebras. The relevant underlying structures are braided algebras and braided coalgebras. We provide some interesting examples of these algebras and coalgebras related to quantum groups. We introduce quantum multi-brace algebras which are generalizations of both braided algebras and $\textbf{B}_\infty$-algebras, as the natural framework. This new subject enables one to quantize some important algebra structures in a uniform way. Particular interesting examples are quantum quasi-shuffle algebras.