Braiding for categorical algebras and crossed modules of algebras I: Associative and Lie algebras

TL;DR

This paper explores the relationships between braided categorical associative and Lie algebras, along with their crossed modules, establishing connections across these algebraic structures.

Contribution

It introduces and analyzes the categories of braided categorical associative algebras and crossed modules, relating them to their Lie algebra counterparts.

Findings

Established equivalences between braided categorical associative and Lie algebras.

Connected braided crossed modules of associative and Lie algebras.

Provided foundational framework for further study of braided algebraic structures.

Abstract

In this paper we study the categories of braided categorical associative algebras and braided crossed modules of associative algebras and we relate these structures with the categories of braided categorical Lie algebras and braided crossed modules of Lie algebras.