Homologies of Algebraic Structures via Braidings and Quantum Shuffles

TL;DR

This paper develops a unified homology framework for various algebraic structures using pre-braidings and quantum shuffles, extending and categorifying existing theories with novel structural insights.

Contribution

It introduces new structural pre-braidings for algebraic structures and a general homology theory based on quantum co-shuffle comultiplication, unifying multiple known theories.

Findings

Generalized homology theories for algebraic structures

Categorification of algebraic homology constructions

Efficient treatment of homology operations using shuffle tools

Abstract

In this paper we construct "structural" pre-braidings characterizing different algebraic structures: a rack, an associative algebra, a Leibniz algebra and their representations. Some of these pre-braidings seem original. On the other hand, we propose a general homology theory for pre-braided vector spaces and braided modules, based on the quantum co-shuffle comultiplication. Applied to the structural pre-braidings above, it gives a generalization and a unification of many known homology theories. All the constructions are categorified, resulting in particular in their super- and co-versions. Loday's hyper-boundaries, as well as certain homology operations are efficiently treated using the "shuffle" tools.