A Class of Bicovariant Differential Calculi on Hopf Algebras

TL;DR

This paper introduces a broad class of bicovariant differential calculi on quantum groups, generalizing existing calculi and connecting to braided group theory, with specific examples on $SL_q(2)$ and quantum groups $A(R)$.

Contribution

It presents a new, extensive class of bicovariant differential calculi on quantum groups linked to $Ad$-invariant elements, expanding the framework for quantum group differential structures.

Findings

Recovers Woronowicz's 4D calculus on $SL_q(2)$

Constructs a sequence of calculi on quantum groups $A(R)$

Connects differential calculi to braided group theory

Abstract

We introduce a large class of bicovariant differential calculi on any quantum group $A$, associated to $Ad$-invariant elements. For example, the deformed trace element on $SL_q(2)$ recovers Woronowicz' $4D_\pm$ calculus. More generally, we obtain a sequence of differential calculi on each quantum group $A(R)$, based on the theory of the corresponding braided groups $B(R)$. Here $R$ is any regular solution of the QYBE.