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Maybe of interest: repairing the definition of hopfish algebra.
N O N C O M M U T A T I V E G E O M E T R Y Seminar of the Mathematical Institute of the Polish Academy of Sciences Ul. Sniadeckich 8, room 322 Monday 11 March 2013 10:15 KENNY DE COMMER (Universite de Cergy-Pontoise) ANTIPODES FOR HOPFISH ALGEBRAS
The notion of a Hopfish algebra was introduced by Tang, Weinstein and Zhu. From the viewpoint of non-commutative geometry, it is a very natural generalization of a Hopf algebra. In contrast with Hopf algebras, a Hopfish algebra has as structural maps bimodules rather than homomorphisms of algebras. Although the bialgebra conditions are easily transferred to this setting, the antipode condition is more problematic. The one imposed by Tang, Weinstein and Zhu is motivated and justified by Poisson geometry, but it seems to lack good properties. In this talk, we want to propose a different notion of antipode which has more structural content. This is work in progress, joint with J. Vercruysse.
I swear to god, “hopfish” sounds like something made up by Dr. Seuss. :-)
I think there is a clear criterion that characterizes the good definition of hopfish algebra, by looking at the Tannaka duality table:
Hopfish algebras should be precisely those sesquialgebras such that their categories of modules are equivalently the rigid monoidal categories.
For that’s what the sesqui/-ishy part is about: all the traditional notions of algebras-with-special properties that appear in the Tannaka duality-table characterize via their modules monoidal categories that have a fiber functor. The point of passing from algebras to sesquialgebras is to get rid of that restriction of having a fiber functor. Since Hopf algebras are characterized by the fact that their algebras of modules are the rigid monoidal categories with fiber functor, their sesqui/-ishy generalization should be such that it gives all rigid monoidal categories.
I’d say.
Todd #3, that thought occurred to me too (Friday 21entry). I also like their comment that it “retains a hint of the Poisson geometry which inspired some of our work”.
Urs #4, I’m sure it would do me a lot of good to work it out myself, but is there still going to be a Yoneda lemma argument for Tannaka duality in situations without a fiber functor?
David,
in the situation without a fiber functor it’s strictly speaking not “Tannaka duality” anymore. So, no (or at least I see no way to bring in Yoneda here, maybe there is).
Maybe a better name than “Tannaka duality” for this more general correspondence between types of (sesqui-)algebras and types of categories is “2-basis theory of 2-modules” or “2-matrix theory”. In the sense of 2-rings and their 2-modules .
The point of passing from algebras to sesquialgebras is to get rid of that restriction of having a fiber functor.
Hm, already the mixed Tannaka duality does not have a (single) fiber functor but rather one deals with various “partial” fiber functors for different extension fields. This stems from the fact that one deals with a gerbe after the reconstruction. Besides, the Tannaka works rather poorly over general rings, with some exceptions like the Nori’s Tannakian theorem. Thus, while very interesting, it is not easy to resolve how to go beyond the known cases even with a seamingly simple principle like the one you suggest. You see, the definition of Hopf algebra works nicely over commutative rings. It would be awkward to have a list of axioms for some generalizations which works only over some rings and not all. I doubt that the definition of Weinstein et al. satisfies the criterion you like to propose in a reasonable generality.
Somehow I had missed that last comment of Zoran above.
To maybe clarify what I was saying, I have written the statement into the entry here: hopfish algebra – Module categories and Tannaka duality.
Are you saying that it is a conjectural class ?
I suppose so far nobody has checked what the module categories of the hopfish algebras as currently defined are, right? And it’s not clear if the definition of the antipode for hopfish algebras is good. Right? I am saying: the sesquialgebra generalization of Hopf algebra clearly must be such that its module categories are rigid monoidal without, necessatrily, a fiber functor. In principle one can turn this around and deduce what the right notion of antipode must be. But I gather nobody has done that yet.
Well, some colleagues tell me that in hopfish algebra there are some axioms without clear categorical meaning so it is not very likely (but I am not competent). On the other hand, nothing tells you that such an algebraic structure exists (and what it means to be a representation! this appeared nontrivial already for Hopf algebroids where even the left and right bialgebroid component do not have necesarilly the same representations).
I’d think the Eilenberg-Watts theorem guarantees that there is a bimodule corresponding to the duality map on a monoidal category, if any. One would just have to characterize it. That’s the antipode bimodule then.
Urs 12: this about the bimodule is an interesting insight!
However, I am still skeptical, as it seems that the lessons of the theory of Hopf algebroids are not taken into account, and they should provide a special case (intermediate between Hopf algebras and whatever you want to achieve, and I would not expect that they are properly included in hopfish algebras so far). Their categories of modules are rigid monoidal categories with fiber functor to the category of bimodules (for bialgebroids skip rigid).
By the way, the weak Hopf algebras are included into Hopf algebroids.
By the way, do you have a reference that states the Tannaka duality for Hopf algebroids precisely?
And what’s a reference for the inclusion of weak Hopf algebras into Hopf algebroids? Also, I was looking for a reference that talks more about the inclusion of weak Hopf algebras into hopfish algebroids.
(All these things are of course fairly straightforwad to work out, but it’s a bit trediuous…)
Look at Hopf algebroid entry: Gabi Bohm’s handbook article listed there has a short account of how weak Hopf algebras are a special case of Hopf algebroids, and also refers to more detailed references (again some are by Gabi). The Tannaka duality for bialgebroids and Hopf algebroids is in the papers by Szlachanyi (on the arXiv) and references there in. I think I listed some of the references at Tannaka duality by Szlachanyi, then a special case by Phung Ho Hai, also Bruguières…
I also mention that some things about characterizing Hopf among bialgebras have more abstract versions and generalizations. One very interesting direction quite much discussed in the community is studied in the paper
Thanks for all the pointers!
While we are at it: do you happen to know references for further refinements:
what’s the status of Hopf-C*-algebra and its Tannaka duality (glancing over the literature seems to reveal several independent and unrelated approaches)
what’s known about climbing higher up the n-algebra-ladder? Next we want to consider algebra objects internal to hopfish/sesqui-algebras, hence certain “tri-algebras”. And so forth. Anyone?
ah, trialgebra
I do not know much about Tannaka in Hopf-C-star setups, further than the original work of Woronowicz.
what’s known about climbing higher up the n-algebra-ladder
Well there are generalizations in A-infinity algebra world, of bialgebras and Hopf algebras, most notably highly combinatorially nontrivial work by Saneblidze.
there are generalizations in A-infinity algebra world, of bialgebras and Hopf algebras
Okay, that’s going to $(\infty,2)$-algebras. But I am wondering about $(\infty,n)$-algebras for $n \geq 3$. Sesquialgebras/hopfish algebras equipped with yet one more algebra structure exhibited by a product sesqui-bimodule, etc. Hence trialgebras, but with the genuine trialgebra structure.
What about Majid’s recent preprint on “strict quantum 2-groups” ? Is this equivalent to a special case of trialgebra ? It would be interesting to do a multiobject version of Majid’s construction, I mean things provided by say crossed modules of Hopf algebroids instead of crossed modules of Hopf algebras…
Hm, maybe. I need to look at this. Thanks for the pointer.
But just as to clarify more what I am after: I have expanded a bit now at trialgebra and at Hopf monoidal category.
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