Determine whether braided monoidal categories systematically alter bicovariance

Determine whether working within a braided monoidal category, compared with the ordinary category of vector spaces, systematically causes a loss or a gain of bicovariant differential calculi and related structural properties.

Background

The paper establishes that bicovariance can be restrictive in braided settings, as illustrated by the non-existence of non-trivial braided bicovariant calculi on the braided quantum plane, but it also presents a transmutation of Sweedler’s Hopf algebra in which a calculus gains braided bicovariance. These examples demonstrate that braided structures can produce either outcome relative to the ordinary vector-space setting.

The authors obtain partial answers for quasitriangular and coquasitriangular Hopf algebras through transmutation, but explicitly state that no general pattern is expected beyond those contexts and that the outcome depends sensitively on the particular case. Thus, the general question of whether braided monoidal categories systematically enhance or restrict bicovariance remains unresolved.