On the braiding on a Hopf algebra in a braided category
The paper is based upon the following simple observation: If \(H\) is a Hopf algebra with an antipode, the compatibility condition of a bialgebra can be solved for the braiding. In particular, the braiding \(\tau_{HH}\colon H\otimes H\to H\otimes H\) is uniquely determined by the algebra and coalgebra structure \(\nabla\) and \(\Delta\) if an antipode \(S\) exists, that is, \(\tau_{HH}=(\nabla\otimes\nabla)(S\otimes\Delta\nabla\otimes S)(\Delta\otimes\Delta)\). This result implies that a monoidal functor between prebraided monoidal categories that maps a Hopf algebra \(H\) to a Hopf algebra, necessarily preserves the braiding on \(H\otimes H\). Several applications are listed. Notably, the observation rules out that any nontrivial examples of commutative (or, cocommutative) Hopf algebras in nonsymmetric braided categories exist. This is a rigorous proof of a version of Majid's observation that commutativity is too restrictive a condition for Hopf algebras in braided categories.
scientific article; zbMATH DE number 1249605