Yetter-Drinfeld-Long bimodules are modules

TL;DR

This paper establishes an isomorphism between the category of Yetter-Drinfeld-Long bimodules over a finite dimensional bialgebra and the Yetter-Drinfeld category over its tensor product with the dual, with a braided isomorphism when H is a Hopf algebra.

Contribution

It proves the categorical isomorphism and braided structure for Yetter-Drinfeld-Long bimodules over finite dimensional bialgebras and their tensor product with duals.

Findings

Category of Yetter-Drinfeld-Long bimodules is isomorphic to that over H⊗H*

Isomorphism is braided if H has a bijective antipode

Provides new structural insights into bimodule categories

Abstract

Let $H$ be a finite dimensional bialgebra. In this paper, we prove that the category of Yetter-Drinfeld-Long bimodules is isomorphic to the Yetter-Drinfeld category over the tensor product bialgebra $H\o H^*$ as monoidal category. Moreover if $H$ is a Hopf algebra with bijective antipode, the isomorphism is braided.