Duality Theorems for Infinite Braided Hopf Algebras

TL;DR

This paper proves duality theorems for infinite-dimensional braided Hopf algebras with symmetric braiding, extending classical duality results to the braided tensor category setting.

Contribution

It establishes the Blattner-Montgomery duality theorem and two additional duality theorems specifically for infinite braided Hopf algebras in Yetter-Drinfeld categories.

Findings

Proves the Blattner-Montgomery duality theorem in braided categories.

Derives duality theorems for infinite braided Hopf algebras.

Extends classical duality results to the braided setting.

Abstract

Let $H$ be an infinite-dimensional braided Hopf algebra and assume that the braiding is symmetric on $H$ and its quasi-dual $H^d$. We prove the Blattner-Montgomery duality theorem, namely we prove $$ (R # H)# H^{d} \cong R \otimes (H # H^{d}) \hbox {as algebras in braided tensor category} {\cal C}.$$ In particular, we present two duality theorems for infinite braided Hopf algebras in the Yetter-Drinfeld module category.