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I am trying to wrap my mind around Tannaka-Krein duality and it seems quite mysterious for me, as well, as its history. So let me ask:

Question: What was the motivation and historical context for works of major contributors to the "Tannaka-Krein theory" (in a broad sense)? Just to name a few names: Tannaka, Krein, Saavedra, Deligne, Milne, Lurie, (it seems Grothendieck should also be in this list(?)).

Let me explain some points in the history which seems to me puzzling:

  1. Mark Krein was a famous Soviet mathematician, but he was an expert in analysis, it seems it is the only paper by him devoted to algebra (See discussion below). How did he come to it? Why did he not continue?

  2. Similar question about Tadao Tannaka. "His interest in mathematics lied mainly in algebraic number theory", And it seems similar to Krein, it is the only work by devoted to group theory. (See his publication list).

  3. P. Deligne seems to have devoted quite much efforts on "Tannakian formalism" and more generally on tensor categories. What was his motivation? He is a leading algebraic geometer. So probably the subject should be quite important in algebraic geometry? What is its importance?

  4. Wikipedia article starts with a sentence: "...natural extension to the non-Abelian case is the Grothendieck duality theory." What is the role of Grothendieck in this history ? And what is "Grothendieck duality theory" - wikipedia links to something not related.

  5. Important work was done by Saavedra. It seems not so much is known about him, his motivation, his other works.

  6. J. Lurie seems to develop the theory further (see e.g. MO question Tannakian formalism). What is the motivation?

List of references (it seems original articles by Tannaka and Krein are not available electronically)

Tadao Tannaka, Über den Dualitätssatz der nichtkommutativen topologischen Gruppen, Tohoku Math. J. 45 (1938), n. 1, 1–12 (project euclid has only Tohoku new series!)

M.G. Krein, A principle of duality for bicompact groups and quadratic block algebras, Doklady AN SSSR 69 (1949), 725–728. in Russian: М. Г. Крейн, Принцип двойственности для бикомпактной группы и квадратной блок-алгебры, Докл. АНСССР, 69:6 (1949), 725–728.

N. Saavedra Rivano, Cat´egories tannakienns, Lecture Notes in Math., vol. 265, Springer-Verlag, Berlin–New York, 1972.

Deligne, P., and Milne, J.S., Tannakian Categories, in Hodge Cycles, Motives, and Shimura Varieties, LNM 900, 1982, pp. 101-228". ( http://www.jmilne.org/math/xnotes/tc.html )

Some remarks about Mark Krein. Part of his publication list is here, strangely enough the paper on "Tanaka-Krein duality" is not contained in this list.

I have found an article devoted to overview of his works related to group theory: L. I. Vainerman. On M. G. Krein's works in the theory of representations and harmonic analysis on topological groups Ukrainian Mathematical Journal 46 (1994), no. 3, 204-218.

It seems he had several papers dating from 1940-1949 which were related to "Tannaka-Krein theory".

He started as student of Nikolai Chebotaryov, who is famous for Chebotarev density theorem, but actually was also working on Lie groups: famous results Ado theorem and Jacobson-Morozov theorem were obtained by his students Igor Ado and Morozov, who worked in Kazan city Russia. But it is not clear whether Krein was influenced by Chebotarev in this respect, since they meet around 1924 in Odessa city, and the paper was written in 1949, when Chebotarev already passed and long before he moved from Odessa to Kazan city, while Krein stayed in Odessa.

Anatoly Vershik in his paper devoted to 100-anniversary of M. Krein suggests that it might be that "success of Gelfand's theory of commutative normed rings" influenced Krein.

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Could you include a reference on Krein's paper? –  Alexandre Eremenko Jan 18 '14 at 15:09
The reference to Grothendieck duality theory in the Wikipedia article can be safely deleted. –  S. Carnahan Jan 18 '14 at 17:14
@S.Carnahan you should do it, if you are confident. –  Alexander Chervov Jan 19 '14 at 14:56
@AlexanderChervov Done! –  S. Carnahan Jan 19 '14 at 15:39
@S.Carnahan Wow! Respect ! –  Alexander Chervov Jan 19 '14 at 17:30

3 Answers 3

(Edited to correct mistakes signaled in comments below).

I don't know much about the first steps on the theory, Krein and Tannaka. I can just say their work answers a question that seems very natural now, and that I think was natural even then. Since the beginning of the 20th century, representations of groups had been studied, used in many part of mathematics (from Number Theory, think of Artin's L-function to mathematical physics) and more and more emphasized as an invaluable tool to study the group themselves. It was therefore natural to see if a group (compact say) was determined by its representations.

But then, I want to insist on the fundamental role played by Grothendieck in the development of the theory. This role comes in two steps. First Grothendieck developed a pretty complete end extremely elegant theory for a different but analog problem: the problem of determining a group (profinite sat) by its category of sets on which it operates continuously. It is what is called "Grothendieck Galois Theory", for Grothendieck did that in the intention of reformulating and generalizing Galois theory, in a way that would contain his theory of the etale fundamental groups of schemes. What Grothendieck did, roughly, was to define an abstract notion of Galois Category. Those Category admit special functors to the category of Finite Sets called Fibre Functors. Grothendieck proves that those functors are all equivalent and that a Galois category is equivalent to the category of finite set with G-action, where G is the group of automorphism of a fibre functor. He then goes on in establishing an equivalence of categories between profinite groups and Galois categories, with a dictionary translating the most important properties of objects and morphisms on each side. This was done in about 1960, and you can still read it in the remarkable original reference, SGA I.

Already at this time, according to his memoir Recoltes et Semailles, Grothendieck was aware of Krein and Tannaka's work, and interested in the common generalization of it and his own to what would become Tannakian category, that is the study of categories that "look like" categories of representations over a field $k$ of a group, As he had many other things on his plate, he didn't work on it immediately, but after a little while gave it to do to a student of him, Saavedra. As Grothendieck was aware, the theory is much more difficult than the theory of Galois categories. Saavedra seems to have struggled a lot with this material, as would have probably done 99.9% of us. He finally defended in 1972, two years after Gothendieck left IHES, and at a time he was occupied by other, in part non-mathematical subject of interest. Saavedra defined a notion of Tannakian category (as a rigid $k$-linear tensor category with a fibre functor to the category of $k'$-vector space, $k'$ being a finite extension of $k$) but he forgot one important condition (then $End(1)=k$) and some of the important theorem he states are false without this condition.

After that, mathematics continued its development and Tannakian categories began to sprout up like mushrooms (e.g. motives (69, more or less forgotten until the end of the 70's), the dreamt-of category of automorphic representation of Langlands (79), to name two extremely important in number theory). Then Milne and Deligne signaled in 1981 the mistake mentioned above in Saavedra's thesis, gave a corrected definition of Tannakian'a category, and was able to prove the desired theorem in the so-called neutral case, when $k'=k$ (I believe with arguments essentially present in Saavedra). Later with serious efforts, Deligne proved those theorem in the general case. Modern theory have added many layers of abstraction on that.

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Thank you very much for your very interesting answer ! If it would be possible to add more comments, why topic is important in algebraic geometry that would be very kind of you. As a small remark on "...and the language of category, which was becoming popular at that time...", it seems neither Tanaka (1938), nor Krein (1949) used the language of categories. Krein seems used something "block-algebras" also called "Krein-algebras". I wonder when the language of categories stared to be used for this topic ? –  Alexander Chervov Jan 19 '14 at 13:57
Some corrections: the error in Saavedra's thesis is where he "proves" that fibre functors are locally isomorphic (not the existence). In fact, with his definition of Tannakian category they aren't (so the fibre functors don't form a gerbe). For this it is necessary to require that End(1)=k. The error was discovered by Deligne, pointed out in Deligne and Milne, and corrected later (with some difficulty) by Deligne. –  abz Jan 19 '14 at 18:33
Dear Alexander and Anon, thanks for your corrections. I'll try to do some research and come up with an improved version of my post, if possible free of mistakes. –  Joël Jan 19 '14 at 19:14
@AlexanderChervov: Indeed, neither Tannaka nor Krein used categories in their contributions. Again, mostly because their viewpoint was that of "classical" abstract harmonic analysis. As mentioned in my answer, it seems that Hochschild was the first to realize that the algebraic structure Krein identified for the dual of a compact group (a "Krein algebra") was actually a Hopf algebra structure. Tannaka-Krein duality can then be rephrased as an equivalence between the (opposite) category of compact groups and a subcategory of the category of Hopf algebras. I don't know who did this last step... –  Pedro Lauridsen Ribeiro Jan 20 '14 at 5:51
For a categorical point of view in an essentially classical framework, an interesting reference is the paper « Gèbres » by J-P. Serre, Ens. Math 39 (1993) p. 33-85. –  ACL Jan 20 '14 at 8:02

If you look at books on (not just Abelian) abstract harmonic analysis, such as Hewitt-Ross, they tell that Tannaka-Krein duality was originally a non-Abelian version for compact topological groups of Pontryagin duality for locally compact Abelian topological groups $G$. The Pontryagin dual $\widehat{G}$ is also (unfortunately, not necessarily locally compact) an Abelian topological group (it consists of all continuous multiplicative characters with pointwise multiplication, endowed with the compact-open topology), and allows one, for instance, to define a notion of Fourier transform in $G$ using its Haar measure (which is just a multiple of Lebesgue measure in case $G=\mathbb{R}^n$). More generally, the whole topic of duality for locally compact topological groups is a blend of algebra and analysis, just as Schwartz's theory of distributions. Since Tannaka-Krein duality was first formulated in this way, this explains the (harmonic) analyst's interest on the topic.

It was noticed later that the same framework for compact Lie groups, formulated in the language of Hopf algebras (I believe this was done for the first time in Hochschild's book "The Structure of Lie Groups". Hewitt-Ross's uses the older terminology "Krein algebras"), could be extended to algebraic groups, so the topic also fits naturally within algebraic geometry. Moreover, since group duality essentially tells us that we can recover the group from its representation theory (i.e. its "dual"), one may think of moving that framework to the context of G-bundles, or, more generally, gerbes and stacks (and to even higher categorical contexts). That's what Deligne, Lurie and other people did, it seems to me.

Just a side remark: independently from Deligne's work, there is also another categorification of Tannaka-Krein duality using C*-algebras (more precisely, tensor C*-categories), concluded more or less at the same time as Deligne (after more than 15 years of hard work) by Doplicher and Roberts, in the context of the algebraic analysis of superselection sectors in quantum field theory. This framework applies to precisely the same context as the original Tannaka-Krein duality, i.e. to compact topological groups.

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An answer only to question 5): Neantro Saavedra was a chilean mathematician who did his Ph.D. at IHES under the direction of Grothendieck - in fact he was the last of Grothendieck's student at IHES (he defended in 1972). I would guess that his motivation was essentially to get his degree, and that the ideas came from Grothendieck. After his thesis he wrote a paper on "Finite geometries in the theory of theta characteristics" in Enseignement Mathématique (1976), then changed subject and obtained a Ph. D. in Economics at Columbia. He has been Professor of Economics at Tsukuba University in Japan, where he is now emeritus.

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"I would guess that his motivation was essentially to get his degree, and that the ideas came from Grothendieck" - I have no idea whether this is true or false, but I think this kind of speculation has the negative effect of reinforcing the star system, whereby the contributions of the famous are inflated and those of relative unknowns are minimized. For all the evidence we have, all the ideas in Saavedra's thesis came from Saavedra, and none from Grothendieck. I'm not saying that's what happened, just that it's unfair to make assumptions based on reputation alone. –  Tom Leinster Jan 18 '14 at 19:08
You are right, my formulation was too harsh. However I think what Joel says is accurate: Grothendieck had all the ideas in his mind, but no time (or no desire) to work out the details, which he proposed to a student. This is what happened with most of his students. –  abx Jan 18 '14 at 19:37
@TomLeinster Grothendieck is extremely severe with Neantro Saavedra in Récoltes et Semailles and explicitly asserts that the thesis contains less than the notes he gave Saavedra at the onset of his work. As this part of the book is fraught with paranoia, innuendoes and borderline psychotic accusations, I am not sure what to conclude, though. –  Olivier Jan 18 '14 at 21:38
What Grothendieck says about Saavedra's thesis in Récoltes et Semailles is so unreliable, I wouldn't dignify it as giving "one side of the story". Deligne, who was effectively Saavedra's adviser for his last two years, has a very different view. I'm prepared to believe that Saavedra proved less than Grothendieck hoped was true, but because the (Grothedieck's?) definition of a Tannakian category was wrong, what Grothendieck hoped probably wasn't actually true. –  abz Jan 19 '14 at 2:01
@anon Yes, I agree, this part of Récoltes et Semailles is pretty remarkable (not in the good sense): on the one hand, G. says that Saavedra was so mediocre that he could not have produced his lecture note, that Deligne probably intentionally let Saavedra's mistake be published in order to reap the glory later and that he himself recalls being aware of the difficulties and solutions before Saavedra's even started his thesis; then a few pages later he admits that he was at Saavedra's PhD defense, did not notice the mistake and that he has no recollection of it all anyway (even of being there). –  Olivier Jan 19 '14 at 9:54

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