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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeApr 8th 2013

The Idea-section at quasi-Hopf algebra had been confused and wrong. I have removed it and written a new one.

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeApr 8th 2013
• (edited Apr 8th 2013)

Why removing it ? You removed the important and more general viewpoint how it came. The case you describe now in the idea section is just a very special case. Drinfel’d worked in much larger generality of rational CFTs than the more special work of Pasquier Roche Dijkgraaf about at the same time (or a bit later). What was wrong ?

To repeat, Drinfel’d considered the monoidal categories in rational CFTs. Often one had a braided monoidal category structure which could be considered as representations of (possibly weak) quasitriangular Hopf algebras. However, possibly changing the associativity coherences which is allowed in this framework allows for more general twists of Hopf algebras where the axioms involve elements coming from Yoneda when twisting the coherences. Those are quasiHopf algebras. Of course, this procedure can be done more generally also in the case not involving quasitriangular structure, but this is beyond the case of Drinfel’d and beyond the more special case of Dijkgraaf et al.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeApr 8th 2013

The previous version started out by mixing up quasi-triangular Hopf with quasi-Hopf. Next it claimed that the representations of a Hopf algebra constitute a braided monoidal category, which is wrong. Then it said that “the Hilbert space is a representation of a Hopf algebra”, which I can’t make sense of. All that should stay removed.

But I guess what you want to see re-installed is the sentence

This framework allows for natural class of weakenings by changing associativity coherence, what amounts to the twist by a nonabelian bialgebra 3-cocycle and yields the notion of a quasi-Hopf algebra as introduced in…

I am now putting that back in.

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeApr 8th 2013
• (edited Apr 8th 2013)

I edited 1 above – it did not mix quasitriangular – this was the case in original motivation for quasiHopf algebras, what includes your case! I think this also needs reinsertion in some form.

“the Hilbert space is a representation of a Hopf algebra”, which I can’t make sense of.

The Hilbert space of the theory is a representation – it belongs to the braided monoidal category mentioned. For example for gauged Wess-Zumino model for su(n) the detailed structure of this representation is investigated in a series of works of Hadjiivanov, Furlan, Stanev and Todorov, and my undergraduate diploma was studying some aspects of this.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeApr 8th 2013
• (edited Apr 8th 2013)

Yeah, I guess the quasi-triangular Hopf/quasi-Hopf mixup I introduced when putting cross-links too hastily.

I have now edited quasi-Hopf algebra. Check if you agree.

What do you mean by “I edied 1 above”? You edited where?

• CommentRowNumber6.
• CommentAuthorzskoda
• CommentTimeApr 8th 2013
• (edited Apr 8th 2013)

I edited my 1st nForum comment above i.e. 2 (you respond so quickly, that you posted 3 and 5 when I was still editing additions to 2 and 4).

• CommentRowNumber7.
• CommentAuthorzskoda
• CommentTimeApr 8th 2013

I edited now the entry, do you agree now ?

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeApr 8th 2013

Thanks! Good. I have slightly edited a bit more (mostly putting in more hyperlinks.)

• CommentRowNumber9.
• CommentAuthorzskoda
• CommentTimeSep 18th 2019

More references, including 2019 monograph by Bulacu et al.

• CommentRowNumber10.
• CommentAuthorzskoda
• CommentTimeOct 3rd 2019

I wrote the definition of a quasibialgebra in much greater detail and the connection to MacLane’s pentagon.

• CommentRowNumber11.
• CommentAuthorzskoda
• CommentTimeOct 3rd 2019

Added a section on twisting of quasibialgebras by $2$-cochains.

• CommentRowNumber12.
• CommentAuthorzskoda
• CommentTimeOct 3rd 2019

It would be interesting to find a higher-categorical explanation of the nonabelian cocycles and cohomology for bialgebras as defined by Drinfeld and Majid. It is a nonabelian cohomology different from one for groups and it specialized to Abelian cohomology of groups and Lie algebras in particular cases. There are nonabelian 3-cocycles which are related to other nonabelian 3-cocycles in examples like quasi-Poisson structures, nonabelian group 3-cohomology, Pasquier-Roche-Dijkgraaf-Witten model etc. Recent nonassociative geometries coming from fluxes in M-theory and nongeometric string backgrounds as studied in these terms more recently by Luest, Szabo and others are also stated via quasi-Hopf algebras and Drinfeld-Majid 3-cocycles/associators.

1. Fixed a typo about the left and right counit corollaries of (iii). This now accurately reflects (3.1.11) from “Quasi-Hopf Algebras: A Categorical Approach” by Bulacu, et al.

Sean Sanford

2. Just fixing minor typos

Olivia Borghi

3. Just fixing minor typos

Olivia Borghi

4. The “suitable notion of antipode” is actually pretty complicated so I wanted to illuminate that.

Olivia Borghi

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeJan 23rd 2021

Thanks!

I have just reinstantiated the term “a suitable kind of antipode” with the formula, to orient the reader. Okay?