R-Matrices, Yetter-Drinfel$'$d Modules and Yang-Baxter Equation

TL;DR

This paper explores the connections between R-matrices and Yetter-Drinfel$'$d modules in solving the Yang-Baxter equation, introduces weak R-matrices, and develops a braided system framework to unify various homology theories.

Contribution

It generalizes the concept of R-matrices to weak R-matrices and constructs a braided system that encodes YD module axioms, unifying multiple homology theories.

Findings

Weak R-matrices extend solutions to the Yang-Baxter equation.

A braided system encodes all YD module axioms.

The homology of YD modules is unified within braided homologies.

Abstract

In the first part we recall two famous sources of solutions to the Yang-Baxter equation -- R-matrices and Yetter-Drinfel$'$d (=YD) modules -- and an interpretation of the former as a particular case of the latter. We show that this result holds true in the more general case of weak R-matrices, introduced here. In the second part we continue exploring the ''braided'' aspects of YD module structure, exhibiting a braided system encoding all the axioms from the definition of YD modules. The functoriality and several generalizations of this construction are studies using the original machinery of YD systems. As consequences, we get a conceptual interpretation of the tensor product structures for YD modules, and a generalization of the deformation cohomology of YD modules. The latter homology theory is thus included into the unifying framework of braided homologies, which contains among others Hochschild, Chevalley-Eilenberg, Gerstenhaber-Schack and quandle homologies.