Classification of bicovariant differential calculi over free orthogonal Hopf algebras

TL;DR

This paper establishes a classification of finite dimensional bicovariant differential calculi over free orthogonal Hopf algebras by leveraging monoidal equivalences with quantum groups like al_q(SL_2).

Contribution

It introduces a classification method for bicovariant differential calculi over free orthogonal Hopf algebras using monoidal equivalences with quantum groups.

Findings

Classified finite dimensional bicovariant differential calculi over al_q(SL_2) for non-root of unity q.

Extended the classification to free orthogonal Hopf algebras via monoidal equivalence.

Established that monoidal equivalence preserves categories of bicovariant differential calculi.

Abstract

We show that if two Hopf algebras are monoidally equivalent, then their categories of bicovariant differential calculi are equivalent. We then classify, for $q \in \mathbb{C}^*$ not a root of unity, the finite dimensional bicovariant differential calculi over the Hopf algebra $\mathcal{O}_q(SL_2)$. Using a monoidal equivalence between free orthogonal Hopf algebras and $\mathcal{O}_q(SL_2)$ for a given $q$, this leads us to the classification of finite dimensional bicovariant differential calculi over free orthogonal Hopf algebras.