Tannaka-Krein duality in Chari-Pressley's book

Question

I am not sure that this was not discussed before, so excuse me in this case. This can be considered as a special case of my previous question here.

V.Chari and A.N.Pressley in their "Guide to Quantum Groups" formulate the following version of Tannaka-Krein's duality (Theorem 5.1.11):

Let $R$ be a commutative ring, let $C$ be an essentially small, $R$-linear, rigid, abelian, monoidal category, and let $\varPhi:C\to\operatorname{mod}_R$ be a $R$-linear exact faithful monoidal functor. Then there exists a Hopf algebra $A$ over $R$ and an equivalence of $R$-linear categories $C\to\operatorname{corep}_A$ whose composite with the forgetful functor $\operatorname{corep}_A\to\operatorname{mod}_R$ is $\varPhi$.

They give an outline of the proof, but I must say that I don't understand some of its details. I think that there must exist an accurate text with complete proof. Can anybody give me a reference to it?

Also if somebody could cast some light on possible generalizations of this proposition (to the case of Hopf algebras in braided monoidal categories), I would appreciate this very much.

Answer 1

I do not know much about the Tannaka-Krein duality itself. But regarding the last part of your question

Also if somebody could cast some light on possible generalizations of this proposition (to the case of Hopf algebras in braided monoidal categories)

i think that bosonization/transmutation and reconstruction methods between hopf algebras and hopf algebras in braided monoidal categories, may be closely related to what you are looking for. Roughly speaking:

The bosonization method consists of the following idea: Given a hopf algebra $A$ in the braided monoidal category $\mathcal{C}$ one can construct an ordinary hopf algebra $\mathcal{A}$ such that the category of the $A$-modules in $\mathcal{C}$ (i.e. the braided modules of $A$) and the category of (ordinary) modules of $\mathcal{A}$ are in a bijective correspondence which provides an equivalence of categories (between the corresponding module categories).

In the case that $\mathcal{C}={}_{H}\mathcal{M}$, where $H$ is some quasitriangular hopf algebra i.e. in the case in which the initial braided monoidal category is the category of representations of some quasitriangular hopf algebra $H$, then it can be shown that the (ordinary) hopf algebra $\mathcal{A}=A\ltimes H$ is a smash product hopf algebra.

A special case of the above has proved particularly interesting for physics: If $H=\mathbb{CZ}_2$ then the hopf algebra $A$ in ${}_{\mathbb{CZ}_2}\mathcal{M}$ is a hopf superalgebra and the smash product hopf algebra $A\ltimes\mathbb{CZ}_2$ is an ordinary hopf algebra with categorically equivalent representation theory to the initial superalgebra.

At the level of physics this can actually be understood in the sense that the hopf superalgebra and the smash product hopf algebra produce the same physics. In other words, SUSY hopf algebras can be substituted by suitably chosen ordinary hopf algebras retaining the same physical interpretation.

The converse of the bosonization technique is the so-called transmutation method. You can find these methods (in their full generality) in: Majid S., Cross Products by Braided Groups and Bosonization, J. of Alg., v. 163, 1, 1994, p.165-190.

On the other hand there are also various reconstruction theories and methods. In the frame of hopf algebras in braided monoidal categories, this amounts to the following idea: Given a braided monoidal category $\mathcal{C}$, we can reconstruct a quastriangular hopf algebra $H$, such that the representation category ${}_{H}\mathcal{M}$ of $H$ is isomorphic to the initial braided monoidal category $\mathcal{C}$.
A nice bibliographical/historical review of this and related ideas can be found in ch. 10, sect. 10.4, p. 201-203 of S. Montgomery's book (Hopf algebras and their action on rings). An example of a (very) simplified reconstruction theorem is provided by theorem 10.4.2, p. 199 of this book.
Finally, i think it would be interesting to have a look at ch. 9.4, p. 467-499, from Majid's book "Foundations of Quantum group theory".

Answer 2

In the case where $k$ is a field this should be worked out in full generality (for Hopf algebras) in Chapter 5 of 'Tensor Categories' by Etingof, Gelaki, Nikshych and Ostrik. This includes the infinite dimensional case.

For the finite dimensional case this can be generalized even further to weak Hopf algebras, so that every Fusion category is monoidally equivalent to the category of representations for a weak Hopf algebra. This is stated in 'On Fusion Categories.' (Corollary 2.22) where it is there referenced as due to Hayashi in 'A canonical Tannaka duality for finite seimisimple tensor categories ' and Szlach´anyi in 'Finite quantum groupoids and inclusions of finite type.' A proof should also be provided in Ostrik's 'Module categories, weak Hopf algebras and modular invariants.' An additional reference is section 7.23 of 1.

As a note, this is referenced in the nlab article, under the table 'Tannaka duality for categories of modules over monoids/associative algebras' for 'Hopf algebras.' Following the Hopf algebras link takes you to the Hopf algebras article which includes a section on 'Tannakian Duality' and includes a link to Bakke's 'Hopf algebras and monoidal categories' and Vercruysse's 'Hopf algebras---Variant notions and reconstruction theorems'. For weak hopf algebras following the links leads to 3 and 5.

Finally, a gold standard reference for understanding Tannakian duality is Pierre Delign's 'Catégories Tannakiennes' and its related generalizations, linked in the nLab article on said paper.

Answer 3

I am baffled by the lack of "classical" (i.e. non-categorical, pre-Saavedra-Deligne) references in the nLab article on Tannaka-Krein duality besides the very earliest ones, including textbook treatments. A complete and self-contained proof of the Tannaka-Krein duality theorem in the language of Hopf algebras can be found e.g. in Section II.3 of the book The Structure of Lie Groups by G. P. Hochschild (Holden-Day, 1965), see Theorem II.3.5, pp. 30. If Hochschild's (excellent but unfortunately long out-of-print) book cannot be found, his proof was reproduced (also in a reasonably self-contained manner) in Appendix 1.B of the book Elements of Noncommutative Geometry by J. M. Gracia-Bondía, J. C. Várilly and H. Figueroa (Birkhäuser, 2001), see Theorems 1.30 and 1.31, pp. 44-45 therein.

Anyhow, Hochschild's proof seems to go back in its essentials to the proof found in his PhD advisor C. Chevalley's classic Theory of Lie Groups (Princeton University Press, 1951), the novelty here being the Hopf algebra language. According to an interview given by Pierre Cartier (who, by the way, extended Tannaka-Krein duality to linear algebraic groups in his paper Dualité de Tannaka des groupes et algèbres de Lie, C. R. Acad. Sci. Paris 242 (1956) 322-325) to Hochschild's former student Walter Ferrer Santos in pp. 1092-1093 of W. F. Santos, M. Moskowitz (eds.), Gerhard Hochschild (1915–2010), Notices of the AMS 58 (2011) 1078-1099, the role of Hopf algebras in Tannaka-Krein duality was suggested by Cartier to Hochschild while visiting the latter at Illinois in the context of the latter's work with Mostow on representative functions of Lie groups in the late 50's. More details on this can also be found in P. Cartier, A Primer of Hopf Algebras, in: P. Cartier, P. Moussa, B. Julia, P. Vanhove (eds.), Frontiers in Number Theory, Physics, and Geometry II: On Conformal Field Theories, Discrete Groups and Renormalization (Springer-Verlag, 2007), pp. 537-615.

There is also a proof in Section 30, Chapter VII of E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Volume II (Springer-Verlag, 1970), but there the older language of "Krein algebras" is used.

A categorical éxposé of the classical Tannaka-Krein duality theorem ("classical" now meaning as formulated by Chevalley and Hochschild, including Cartier's extension to linear algebraic groups), also missing in its nLab article, can also be found in J.-P. Serre, Gèbres, Ens. Math. 39 (1993) 33-85, which on its turn reproduces Rédaction Bourbaki no. 518 (1968).

(fun fact 1: as of now, the Archives Bourbaki only has Rédactions up to 400, so one really needs to go to Ens. Math. for Serre's)

(fun fact 2: here the term "gèbres" is used by Serre to commonly refer to algebras, co(al)gebras and bi(al)gebras in Bourbakistic fashion, it has nothing to do with "gerbes")