New York Journal of Mathematics
Volume 4 (1998) 259-263


Peter Schauenburg

On the Braiding on a Hopf Algebra in a Braided Category

Published: December 4, 1998
Keywords: braided monoidal category, symmetric monoidal category, Hopf algebra, quantum group, braided group, quantum commutativity, braided commutativity
Subject: 16W30, 18D10, 18D35, 17B37

By definition, a bialgebra H in a braided monoidal category (C,τ) is an algebra and coalgebra whose multiplication and comultiplication (and unit and counit) are compatible; the compatibility condition involves the braiding τ.

The present paper is based upon the following simple observation: If H is a Hopf algebra, that is, if an antipode exists, then the compatibility condition of a bialgebra can be solved for the braiding. In particular, the braiding

τHH:H⊗ H→ H⊗ H
is uniquely determined by the algebra and coalgebra structure, if an antipode exists. (The notions of algebra and coalgebra (and antipode) need only the monoidal category structure of C.)

We list several applications. Notably, our observation rules out that any nontrivial examples of commutative (or cocommutative) Hopf algebras in non-symmetric 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.

Author information

Mathematisches Institut der Universität München, Theresienstr. 39, 80333 München, Germany