Fusion in the entwined category of Yetter--Drinfeld modules of a rank-1 Nichols algebra

TL;DR

This paper derives a nonsemisimple fusion algebra from a Nichols algebra within the context of Yetter-Drinfeld modules, suggesting an equivalence with the triplet W-algebra in certain logarithmic conformal field theories.

Contribution

It introduces a new perspective on the fusion algebra via Nichols algebras and explores the structure of Yetter-Drinfeld modules as an entwined category, linking algebraic and conformal field theory models.

Findings

Derived a nonsemisimple fusion algebra from Nichols algebra

Decomposed products of simple Yetter-Drinfeld modules

Suggested equivalence with triplet W-algebra in logarithmic models

Abstract

We rederive a popular nonsemisimple fusion algebra in the braided context, from a Nichols algebra. Together with the decomposition that we find for the product of simple Yetter-Drinfeld modules, this strongly suggests that the relevant Nichols algebra furnishes an equivalence with the triplet W-algebra in the (p,1) logarithmic models of conformal field theory. For this, the category of Yetter-Drinfeld modules is to be regarded as an \textit{entwined} category (the one with monodromy, but not with braiding).