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I gave Drinfel’d double an Idea-section.
Also moved a paragraph on module categories from the References to a Properties-section.
I have an objection against moving the center construction to properties. Namely, the center of a monoidal category is really the idea of the Drinfel’d double, from the modern perspective. So if a Hopf algebra is obtained by the Tannaka reconstruction from some monoidal category, then the Drinfel’d double is the reconstruction from the center of that monoidal category. This perspective (by Majid) also enables to do beyond the finite-dimensional case.
So you want to have it in the Definition-section then? Fine with me. But it was in the References section before! :-)
Never mind, once I am back to Majid’s book I will edited properly to have both generalities (and the dual version, also called Drinfel’d double).
I have completed publication data for this item:
Am hereby removing the following lines which were sitting below this item:
Essentially the same conclusion (with similar motivation and in terms of equivariant fibres of monoidal fibered categories with orbifolds encoded as internal groupoid objects in the base) was independently obtained by Zoran Škoda at MPI Bonn in 2004 (unpublished).
Because, such kind of comment may have the opposite of the desired effect, unless some minimum of reference is provided. Is there maybe an MPI seminar talk title that we could point to, or something similar?
I have fixed and completed the bib-data for the following two items:
Robbert Dijkgraaf, Vincent Pasquier, Philippe Roche, QuasiHopf algebras, group cohomology and orbifold models, Nucl. Phys. B Proc. Suppl 18 (1990) 60-72 [doi:10.1016/0920-5632(91)90123-V]
Robbert Dijkgraaf, Vincent Pasquier, Philippe Roche, Quasi-quantum groups related to orbifold models, International Colloquium on Modern Quantum Field Theory (Bombay, 1990), 375-383, World Sci. (1991) [cds:206306, pdf]
expanded out the publication data of this item:
Vladimir Drinfeld, Quantum groups In A. Gleason (ed.) Proceedings of the 1986 International Congress of Mathematics 1 (1987) 798-820,
expanded version:
Journal of Soviet Mathematics 41 (1988) 898–915 [doi:10.1007/BF01247086]
I have fixed and expanded the publication data for this item:
I have tried to find an original reference on the oft-quoted but rarely (never?) cited fact that the Drinfeld double of a finite-group algebra is the convolution algebra of its inertia orbifold.
The best I could find so far:
(this is made almost explicit in Dijkgraaf, Pasquier & Roche (1990), eq. 2.1.10; the categorified version of this statement is highlighted in Hinich (2007)).
Urs 5: The statement of the theorem is stated in my proposal to IHES submitted much earlier – on Jan 3, 2003 (I do not know how proposals are citable in your opinion). Proposal has been accepted by the IHES expert committee 2 weeks later. Part of proposal had some ideas related to the Baranovsky’s result that additively the periodic cyclic cohomology for the sheaves over orbifold is the same as the orbifold cohomology which had the effect (as discovered by Ruan and Chen) of twisted sectors that is inertia orbifold construction. It was mentioned in the Hinich’s paper that this was one of his motivations as well (not Baranovsky’s paper but the explanation of orbifold cohomology). Bressler’s ideas on cyclic cohomology and groupoids (which I learned from him at a meeting in Arizona in November 2002) related the two.
At the time of the proposal I thought that the theorem was a triviality (but was wrong). My abstract version I actually proved only in winter 2004/2005. Immediately when I proved it (it was 5 am) the same day later I wrote an email letter to Paul Bressler and he responded immediately that Hinich had told him in the Summer (that is 2004) that he proved such a theorem. Bressler commented in the email that this story fits nice (with the rest of the story Bressler on cyclic cohomology was pushing from 2002, you may remember I gave you at some point scan of 17-page part of his informal preprint from April 2004 reflecting some of his ideas, he also gave a talk on this at Mittag Leffler Institute conference NOG organized by Kontsevich). So, with the Bressler’s statement on his conversation with Hinich, I gave up the idea to write already known theorem. However in 2005, when Hinich’s preprint finally appeared on the arXiv I saw that the version of the theorem is slightly different and the proofs quite different so I regretted not to have written an independent paper when the research was fresh with all ideas. Bressler had a number of ideas how the inertia orbifold (and iterated inertia functors) produced important constructions (say modular group actions, Kac-Moody level k representations etc.) and how this is related to cyclic cohomology and my IHES proposal reflected this philosophy with the idea that in the spirit of noncommutative geometry the orbifold has to be replaced by a category of sheaves or quasicoherent sheavesand inertia orbifold is a cyclic version of it. E.g. a theorem of Bressler (motivated by another theorem of Goodwillie) said that the simplicial nerve of inertia groupoid (with its canonical cyclic structure) is isomorphic to the cyclic nerve of the original groupoid. So, once I had the theorem a la Hinich, I concluded that the cyclic cohomology with coefficients could be obtained by forcing Yang-Baxter relations. I investigated this for several days and communicated this to Connes, whom I shown my preprint, which a bit later appeared as https://arxiv.org/abs/math/0412001 (never finished) and he did not like it as it was category theory he said, but he encouraged me to try to modify the construction to get the Hopf cyclic cohomology with coefficients which was in fact my original motivation which I did not tell him. Comonadic resolution had to be cotensored to get this but I did not yet know how to do it (to obtain para-cyclic module in this generalization). I saw that in my preprint the distributive laws played a role but did not make a decisive step to consider them in relative version. This has been resolved in the paper by Bohm and Stefan 2-3 years later https://arxiv.org/abs/0705.3190 and my main result from 0412 is just a special case of a theorem in their paper. I felt ashamed when posting 0412 that I did not fullfill original wish (to get cyclic cohomology with coefficients and possibly some connection to orbifold cohomology) so I skipped the motivation part from the paper and just wrote a couple of formally abstract theorems/constructions with YB relations, distributive laws and para-cyclic objects. There is also a trivial example of cyclic structure there (Karoubi pointed to me that it probably by Dold-Kan corresponds to $(-1)^n$ automorphism and he was right) which is related to some constructions in descent theory by Phillipe Nuss and also by Menini and Stefan (the version of my paper at the arXiv has a trivial error in the description of an algorithm there for description of the $n$-th level $\tau$ for that case).
At MPI I did not give a seminar on this theorem but communicated it only to Lyubashenko there (and it was the subject of numerous conversations with M. Jibladze who explained to me relations to some work of Pirashvili and wrote a computer program for one related constructon of mine). V. Lyubashenko also had an unpublished interesting work on cyclic cohomology at the time, a preprint which he gave me that winter (I lost the file later but one can ask him I suppose, he is in Kyiv now I guess), and which never appeared in public. I only later gave a conference exposition of my results on May 23, 2007 at ICTP, in a talk “Yang-Baxter condition, equivariance and cyclic cohomology”, School and conference on algebraic K-theory and its applications, ICTP Trieste, Italy, May 14-June 1, 2007. The very evening before the talk, Bohm and Stefan had posted a preprint on the arXiv solving cyclic cohomology with coefficients problem which was partly attacked by my study of cyclic cohomology at the time so the talk and large part of the unfinished preprint 0412 became even more obsolete (though the talk was liked by Paul Baum what was making me happy). However, Kaledin’s paper on coefficients for cyclic cohomology from 2007 or so has also at one place a Grothendieck fibration with similar braiding. I persuaded (in 2006 or so) late Rosenberg to look at Kaledin’s paper and allegedly he discovered a nonabelian category which is universal target for cyclic cohomology with coefficients which is hence double universal (nonabelian derived functor is universal for fixed target and now in addition you take universal among all targets). Rosenberg claimed that his discovery was obvious if one reads Kaledin and dares to go nonabelian (in his sense of nonabelian homological algebra via satellites in the presence of subcanonical singleton Grothendieck topology which he calls right ’exact’ structure). He was supposed to give a talk on this at the same conference in Trieste but changed his mind and gave other talks on nonabelian homological algebra. I have access to his manuscripts provided by his son, but in my overview of it there is no manuscript on the “doubly universal cyclic cohomology” and therefore I consider this work lost for us (he was promising me also personal lecture on the construction but I found it impolite to insist to do it “now” until it was too late).
All together I spent some 8 months thinking on the relation of cyclic cohomology and Yang-Baxter phenomena. But all together my results were too partial and reflecting just part of the wider story envisioned by other mentioned people above.
such kind of comment may have the opposite of the desired effect
There is no desired effect here apart from exposing, encouraging and researching into the wider story hinted to above (Hinich’s work is not mentioning many of the motivations and extensions though I am sure he is aware of them; I met him just few days ago in Luxembourg for the first time since winter 2003/2004 and it did not come to my mind to discuss this subject (!), we never discussed it, I only mentioned it in an email message in 2005 or 2006). For example, Baranovsky’s work has proof just in a partial case (say, no proof if you add a gerbe, which plays major role not only in orbifold cohomology but also in Bressler’s story), then resurrecting the Rosenberg’s construction, the connection to the story on adjunctions for cyclic category pointed out by Jibladze, connection to more recent work by Kasangian and Lack etc.
Urs 9: maybe you could ask Pjotr Hajac. I was at a conference at Banach Center in 2001 (organized by Hajac) and I remember somebody gave a talk on the theorem of P.M. Hajac, M. Khalkhali, B. Rangipour, and Y. Sommerhaeuser on the first construction of Hopf cyclic cohomology with coefficients. I remember he was mentioning many times in conversations as well known observation that the equivariant modules for the action by conjugation are simply Yetter-Drinfeld modules and as it is well known that the category of YD modules are the center of the monoidal category of all modules (under finiteness conditions at least) – that is this classical case which you point out (at the level of module categories, the special case of the underlying algebra also there).
This remark of Hajac partially made me think when I was writing the proposal in Dec 2002 to IHES that the orbifold generalization is trivial and Bressler has convinced me that the case for orbifold and transgressions at the level of loop groups are of fundamental importance. If I understand your question, Hajac should be able to tell you the context from the point of view of Hopf algebraists in 1990s (or I am misunderstanding your reference question?).
In fact for the Hopf cyclic cohomology with coefficients stable __anti__Yetter-Drinfeld modules are used, but if the Hopf algebra is just a group algebra then there is no difference. There are later some interpretations of “stable aYD” in terms of connections if I remember right,
but I never understood their arguments. On the other hand, it is important now to me. Namely, Niels Kowalzig has a number of constructions of Gerstenhaber algebras and Batalin-Vilkovisky modules over them (noncommutative differential calculus a la Tsygan-Tamarkin) constructed in terms of bi/Hopf algebroids and aYD modules. These are responsible for such structures on some cases of Hochschild cohomology and some other examples. Luc Menichi gave a talk in Bonn on one of those structures found by hand in 2004 which was inspiring to my study of the subject. Kowalzig later generalized his construction terms of operads and bialgebroids give just some special cases of the more general construction. The appearance of aYD modules in his examples (and corresponding para-cyclic modules like in work of Bohm-Stefan etc.) is interesting from my point of view. As far as bialgebroid examples which I studied in last 8 years or so, those are related to some differential operators in this or other was and also to Yetter-Drinfeld module algebras (not anti). I do not know if the examples of bialgebroids I understand then could be of any use in the further study. But I know enough on Hopf algebroids (mainstream versions) to delve into the subject.
10: Some (but not all) of the ideas from Bressler’s 2004 talk “Levels and characters” at Mittag-Leffler and his preprint from 2002-2004 mentioned in 10 are rediscovered partly by Simon Willerton and Nora Ganther in their papers 3-4 years later. My attempts to persuade Bressler to publish it at the time were unsuccessful as he considered his observations about known to the experts (“folklore”).
To me an important idea to see the fibered category picture of the relation between the Drinfeld center and inertia orbifold was also partly from the observation I learned from Lunts in Spring 2002 that Hopf modules are to be thought as descent data and equivalent (co)simplicial objects in full parallel to equivariant sheaves (for explanation see my paper in Georgian Math. J., https://arxiv.org/abs/0811.4770). The same holds for Yetter-Drinfeld modules (coring picture of all and more generally of Doi-Kopinnen modules discovered by Brzezinski around that time is equivalent to this, though more algebraic). This gave us a tool to derive double complex for Ext-groups for Yetter-Drinfeld modules which is postulated somewhat ad hoc in the paper
Coring picture allows to generalize these arguments to more general mixed distributive laws (as in D. H. van Osdol, Bicohomology theory, Trans. Amer. Math. Soc. 183 (1973), 449–476.). An argument mimicking the one from a lemma in an appendix of the paper https://arxiv.org/abs/0907.3335 by B. Shoikhet (which he attributes to B. Keller) in allows to show directly that it computes Ext groups as well, without referral to a descent like picture.
The notion of equivariant sheaves in noncommutative geometry as studied by Lunts and me and in coring picture by Brzezinski and others have been opposed by another version in a recent paper
The two notions coincide in the commutative case when the Hopf algebra is replaced by a group. Gabi Bohm wrote a MathSciNet review of their paper and discovered (and wrote in the review) a picture in which the construction of D’Andrea and De Paris is related to a comonad which is totally symmetric to the role of a comonad in a work of Lunts and me (sketched in my Georg. Math. J. paper). This is beautiful and it was quite shocking to me, as D’Andrea and De Paris argued that certain map in our construction does not need to be invertible and therefore the theory can be modified using this observation. So I considered their picture as some sort of weakening of ours. But Gabi has shown beautifully that the picture is rather symmetric when stated in different, more abstract, terms.
For me, the study of equivariant bundles in noncommutative geometry was important when I was writing my thesis between 1997 and 2001, where I studied principal bundles over quantum flag varieties and associated equivariant line bundles which would play role in geometric version of quantum Borel-Weil theorem, which partly motivated this study (my eventual goal was to understand coherent states for quantum groups but in a way this is just another point of view on the same subject). Considering the quasicoherent sheaves over these bundles became important only when I had to justify that the meaning of my constructions does not depend on a choice of a localization cover used in the construction. This I understood fully only after the thesis was completed when I (forced by that question) gradually shifted to the study of quasicoherent sheaves, including Hopf-equivariant ones. Only then I understood that my construction have proper treatment in terms of a variant of the approach to noncommutative schemes formulated by Rosenberg.
The category of modules over a quantum double of a finite-dimensional Hopf algebra $H$ is equivalent to the category of Yetter-Drinfeld modules of $H$.
—added “finite-dimensional”
Edit: Yetter-Drinfeld modules have 4 variants – left-left, left-right, right-left and right-right. The categories of YD modules are mutually equivalent if the antipode is invertible (automatic in finite-dimensional case) but not in general. See the classical paper on YD modules for bialgebras by Radford, hence in those cases it does not matter which one we talk about.
The statement of the theorem is stated in my proposal to IHES submitted much earlier – on Jan 3, 2003 (I do not know how proposals are citable
Like this:
* Zoran Škoda *ProposalTitle*, IHES (Jan 2003) [[link](url)]
Thanks (but never mind, the citation is of no importance, unlike the ideas above and their extensions for crepant resolutions). (By “how” I meant “how reasonable”).
The category of modules over a quantum double of a finite-dimensional Hopf algebra $H$ is equivalent to the category of Yetter-Drinfeld modules of $H$.
—added “finite-dimensional”
Anon
On representation theory of the quantum double of a finite group
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