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have created an entry Khovanov homology, so far containing only some references and a little paragraph on the recent advances in identifying the corresponding TQFT. I have also posted this to the $n$Café here, hoping that others feel inspired to work on expanding this entry
I added lots of links to the bibliography (doi, MR…). There was an earlier $n$Café post about a conference on categorification of the type which includes Khovanov’s work – categorification related to quantum groups, knots and links – the one in Glasgow http://golem.ph.utexas.edu/category/2008/11/categorification_in_glasgow.html, the report on Kamnitzer’s work http://golem.ph.utexas.edu/category/2009/04/kamnitzer_on_categorifying_tan.html, as well as link homology post http://golem.ph.utexas.edu/category/2009/02/link_homology_in_paris.html which you may want to link in your intro text on the $n$Café. Edit: I added the links to $n$Café to Khovanov homology.
thanks, Zoran!!
Important further aspects that are missing:
at least a rough account of the definition and some properties
at least a rough mention of the work by Bar-Natan and Lauda on categorified quantum groups and their relation to Khovanov homology.
Myself, unfortunately I don’t have time right now. But maybe somebody reading this here does.
Why do you think that newer work on categorification of quantum groups is closer to Khovanov’s homology than original work from 1990s ?
Why do you think that I think that?
Well, Lauda’s work on the subject is very recent. While recent work of many authors is gradually clarifying the true picture, the categorification of quantum groups was in bloom already 10+ years ago, pushed by e.g. Kazhdan, Igor Frenkel, Bernstein and others. Some of it is documented, some is not much published (like work of Kazhdan); by 2003 when I was leaving US it was rather frequent topic in quantum group/representation theory community; in 2004 I heard a strong talk of Stroppel about it in Warwick, she was pushing the program very enthusiastically (what continues now). I anticipate that you do have some insight why you consider also the young work of Lauda “important” missing aspect and you want to share the motivation with us before we invest time into it unmotivated…
I anticipate that you do have some insight why you consider also the young work of Lauda “important”
Gee, Zoran, all I said is that I think we should list it. Don’t try to fight with me over things I didn’t say. If you have insights on Khovanov homology that you want to share, just add them to the entry.
I have no insights on Khovanov homology (frankly no much interest either), I was working on the entry just to help you. I do have some insights on the categorification of quantum groups, but do not know which of that work is directly relevant to Khovanov homology.
Don’t try to fight with me over things I didn’t say.
I do not understand why asking you for your insight for “Lauda’s work” (in 4 an 6) is “fight”. It was just a minor curiosity (which I mainly lost after 7). Edit: I really (and naturally) expect that if somebody labels something as an important further aspect (as in 3) that one will be happy to hear and answer to the interest why (s)he thinks so.
I added few random related references from Khovanov and from Bar-Natan.
Hey Zoran, Lauda is a collaborator of the very Khovanov and from what i hear people say in the hallways, his work is being well recognized. That’s why I thought he needs to be cited in the entry on Khovanov homology. But that’s about all I know about it.
If you have further insights that for some reason Lauda’s work should not be cited because it is far too little “important” as you suggest in 6, then please share your knowledge.
Thanks for clarifications.
As I explained in detail in 6, as far as categorification of quantum groups is concerned the Lauda’s work is indeed very late. But I expected in 4 that there are arguments “that newer work on categorification of quantum groups is closer to Khovanov’s homology than original work from 1990s”. This is not in contradiction with what you are saying and is not suggesting that it is not important – enhancing existing subject by finding deep connections to other areas is important; I am suggesting that one of its merits is probably in getting closer to Khovanov’s work and was asking for insight in this direction; there may be more (e.g. in making the constructions more explicit). I daresay that quantum groups live per se and the depth of research there is not necessarily measured only by connections to fancier and more popular subjects like TQFT; in an entry on TQFT one should of course emphasise such aspects, with a reference that the subject was largely developed before.
All right, I guess we have clarified then whatever needed clarification.
Phew, the comment #3 was posted when i was in a rush as a reminder because I didn’t (and don’t) have the time to look into all this more closely. My plan that it would save me time to post this reminder did not quite work out ,-)
Okay, need to get back to work.
Sorry, it came that way. On the other hand, when writing to do list one can just say (neutral and short) “to do list” :) as a precaution :)
By the way, in 2002 many experts in Lie theory somewhat expected Nakajima to get a Fields medal for his work on quantum groups and ALE spaces, extending earlier Lusztig’s work. For example, Armand Borel expected that. On the other hand I also heard the opinion, that it is still incomparable to the bulk of deep Lusztig’s work in Lie and representation theory, not only quantum groups, canonical basis and Kazhdan-Lusztig-Deodhar theory; but Lusztig was unfortunately never granted.
Okay, I see, my “important” caused all the toruble.
No problem. Thanks for all your help! Maybe I shouldn’t open new entries when I don’t have enough time. On the other hand, precisely if I don’t have enough time do I feel the need to quickly note down some things in some entry.
You know, you are so nice, literate, willing to explain to people. So even in a hurry you write long sentences full of niceties like “This was nicely explained in ref 2 which took some ideas from ref 4 and build on ref 5”.
But there may be soon another big step in quantum groups from topological point of view. I mean there is some unavailable work in progress from Gaitsgory and Lurie explaining the origin of quantum groups from the point of view of homotopy/higher category theory…I read some enticing abstract on this…
there is some unavailable work in progress from Gaitsgory and Lurie explaining the origin of quantum groups from the point of view of homotopy/higher category theory.
This I would like to see! Do you have any further information?
I know that it is about corollaries of their unfinished work on from this talk by Lurie:
Defining Algebraic Groups over the Sphere Spectrum I
Let G be a compact Lie group. Then the complexification of G can be realized as the set of points of an algebraic group defined over the complex numbers. Better still, this algebraic group can be defined over the ring Z of integers. In the setting of derived algebraic geometry, one can ask whether this group is defined “over the sphere spectrum”. In these talks, I’ll explain the meaning of this question and describe how it can be addressed using recent joint work with Dennis Gaitsgory.
and in
I should note that Lurie already did some work with Gaitsgory on quantum groups, which appeared in (Lurie did not sign as an official coauthor at the end as explained inside)
where a version of the Kazhdan-Lusztig correspondence involving quantum group representations and certain categories of sheaves related to conformal field theory (and affine Lie algebras) is raised to a certain derived picture (eventually equivalence of certain stable (infinity,1)-categories). This is a spectacular further step after earlier work of Finkelberg, Bezrukavnikov and others. David-Ben Zvi commented on MO:
One awesome idea is the use of the $E_2$ perspective to explain WHY quantum groups relate to local systems on configuration spaces of points (Drinfeld-Kohno theorem and its generalizations) and in fact use it to prove the Kazhdan-Lusztig equivalence between quantum groups and affine Lie algebras.
He also quotes that the Lurie’s talk on Koszul duality is relevant (pdf). I guess part of these last things is now in Higher Algebra. Cf. also Carnahan’s comment at MO (and this one) relating the Drinfeld double in the light of Lurie-Gaitsgory work.
That’s great, thanks!
added these pointers:
Mikhail Khovanov, Louis-Hadrien Robert, Foam evaluation and Kronheimer–Mrowka theories, Advances in Mathematics 376 (2021) 107433 [arXiv:1808.09662, doi:10.1016/j.aim.2020.107433]
Mikhail Khovanov, Louis-Hadrien Robert, Conical $SL(3)$ foams, Journal of Combinatorial Algebra 6 1/2 (2022) 79-108 [arXiv:2011.11077, doi:10.4171/jca/61]
Mikhail Khovanov, Universal construction, foams and link homology, lecture series at QFT and Cobordism, CQTS (Mar 2023) [web]
added these poitners:
Nitu Kitchloo, Symmetry Breaking and Link Homologies I [arXiv:1910.07443]
Nitu Kitchloo, Symmetry Breaking and Link Homologies II [arXiv:1910.07444]
Nitu Kitchloo, Symmetry Breaking and Link Homologies III [arXiv:1910.07516]
Nitu Kitchloo, Symmetry breaking and homotopy types for link homologies, talk at QFT and Cobordism, CQTS (Mar 2023) [web]
added pointer to:
Khaled Qazaqzeh, Nafaa Chbili, On Khovanov Homology of Quasi-Alternating Links, Mediterr. J. Math. 19 104 (2022) [arXiv:2009.08624, doi:10.1007/s00009-022-02006-5]
Nafaa Chbili, Quasi-alternating links, Examples and obstructions, talk at QFT and Cobordism, CQTS (Mar 2023) [web]
added more of the original references on the homology of the cobordism category of “foams”:
Mikhail Khovanov, $sl(3)$ link homology, Algebr. Geom. Topol. 4 (2004) 1045-1081 [arXiv:math/0304375, doi:10.2140/agt.2004.4.1045]
Marco Mackaay, Pedro Vaz, The universal $sl_3$-link homology, Algebr. Geom. Topol. 7 (2007) 1135-1169 [doi:10.2140/agt.2007.7.1135]
Marco Mackaay, Pedro Vaz, The foam and the matrix factorization $sl_3$ link homologies are equivalent, Algebr. Geom. Topol. 8 (2008) 309-342 [arXiv:0710.0771, doi:10.2140/agt.2008.8.309]
Marco Mackaay, Pedro Vaz, The diagrammatic Soergel category and $sl(N)$-foams, for $N \gt 3$ [arXiv:0911.2485]
pointer
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