A while ago, I was reading Majid's book Foundations of quantum group theory, and Section 9.4 has a rather fascinating description of a Tannaka-Krein reconstruction result for quantum groups. In particular, there seems to be the claim that if $H$ is a quasi-triangulated quasi-Hopf algebra, then the braided endomorphisms of the identity functor on (a suitably large category of) $H$-comodules form a Hopf algebra object $H'$ in $H$-comodules, in a way that identifies $H'$-comodules with $H$-comodules. Furthermore, $H'$ is commutative and cocommutative with respect to the braided structure on the category of $H$-comodules, and under some nondegeneracy assumptions, it is self-dual. This is called "transmutation" because $H'$ appears to have nicer properties than $H$ (although it may live in a strange category). Some examples are given, e.g., $U_q(g)$ and quantum doubles of finite groups. Unfortunately, the arguments in the proof are given in a diagrammatic language that I was unable to fathom.

Why does this result seem problematic?

The first problem comes from reasoning by analogy. If I want to describe a Hopf algebra object in a monoidal category, I need some kind of commutor transformation $V \otimes W \to W \otimes V$ to even express the compatibility between multiplication and comultiplication, e.g., that comultiplication is an algebra map. In operad language, I need (something resembling) an E[2]-structure on the category to describe compatible E[1]-algebra and E[1]-coalgebra structures. If you think of the spaces in the E[k] operad as configurations of points in $\mathbb{R}^k$, this is roughly saying that you need two dimensions to describe compatible one-dimensional operations. In the above case, the category of $H$-comodules has an E[2]-structure, but I'm supposed to get compatible E[2]-algebra and E[2]-coalgebra structures. Naively, I would expect an E[4]-category to be necessary to make sense of this, but I was unable to wrestle with this successfully.

The second problem comes from a construction I've heard people call Koszul duality, or maybe just Bar and coBar. If we are working in an E[n]-category for n sufficiently large (like infinity, for the symmetric case), then there is a "Bar" operation that takes Hopf algebras with compatible E[m+1]-algebra and E[k]-coalgebra structures, and produces Hopf algebras with compatible E[m]-algebra and E[k+1]-coalgebra structures. There is a "coBar" operation that does the reverse, and under some conditions that I don't understand, composing coBar with Bar (or vice versa) is weakly equivalent to the identity functor. In the above case, I could try to apply Bar to $H'$, but the result cannot have an E[3]-coalgebra structure, since E[3] doesn't act on the category. Applying Bar then coBar implies the coalgebra structure on $H'$ is a priori only E[1], and applying coBar then Bar implies the algebra structure on $H'$ is a priori only E[1]. It is conceivable (in my brain) that the E[2]-structures could somehow appear spontaneously, but that seems a little bizarre.

Question

Am I talking nonsense, or is there a real problem here? (or both?)

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Naturally, you are welcome to replace every appearance of "Hopf" with "bi" if it makes you more comfortable. –  S. Carnahan Jan 31 '10 at 19:54
I don't know the operad language enough to be able to make a stab at answering your questions. I do know that Majid has some notions of "braided Hopf algebra" that make sense in a braided monoidal category. They are somewhat surprising, because they are fairly asymmetric. This is more of a problem for Majid's "braided Lie algebra"s. –  Theo Johnson-Freyd Jan 31 '10 at 21:24

Scott, I believe the source of your confusion is that Majid doesn't claim that the braided Hopf algebra he constructs is both braided commutative and braided co-commutative in C. Just as in the usual case, the Hopf algebra one constructs is braided co-commutative (like U(g)) or braided commutative (like O(G)) if you work dually, but not both. I think your arithmetic about E[n]-operads is correct as to why that would be unusual.

Can you point where in Chapter 9.4 is it claimed that the resulting Hopf algebra is both commutative and co-commutative? For instance in my copy on page 481, he explains U(C) is braided co-commutative but doesn't mention commutative anywhere. On page 477 of my copy, he says "One can use the term 'braided group' more strictly to apply to braided-Hopf algebras which are 'braided-commutative' or 'braided-cocommutative' in some sense." (keyword "or").

By the way, in the Section 3 of http://arxiv.org/abs/0908.3013 is an exposition (not original) reconstructing the algebra A (transmutation of O_q(G)), which uses language probably more familiar to you. Also there is a Remark 3.3 (explained to us by P. Etingof) which gives a more concise description of A (and can be used to derive its key properties like braided-commutativity) by considering module categories and internal homs.