 

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 


Abstract
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 τ_{HH}: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 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.


Author information
Mathematisches Institut der Universität München, Theresienstr. 39, 80333 München, Germany
schauen@rz.mathematik.unimuenchen.de
http://www.mathematik.unimuenchen.de/personen/schauenburg.html

