# Tannaka–Krein duality

First I would like to stress that maybe I don't have a necessary background from the theory of Lie groups. I met the topic of Tannaka–Krein duality while reading the book of Gracia–Bondia, Varilly and Figureoa "Elements of noncommutative geometry". This theory resembles the Gelfand–Najmark theory for commutative $C^*$-algebras but there are some points which are different.

I will follow the aproach from this book. The setting is the following: they start with the Hopf algebra $H$ over $\mathbb{R}$ and consider the set of all algebra homomorphisms $\mathcal{G}(H)$. This set could be equipped with the so called convolution product and it turns out to be a group. One can define the topology of pointwise convergence on $\mathcal{G}(H)$ and $\mathcal{G}(H)$ becomes a topological group. Then a suitable functional $J$ is introduced: as far as I understood, this is a prototype of a Haar integral. It must satisfy certain axioms: a commutative Hopf algebra together with this functional is called a commutative skewgroup. With this extra structure on $H$ it it shown that $\mathcal{G}(H)$ becomes a compact group.

Then comes a theorem concerning Hopf algebras: starting from the commutative skewgroup $H$ one can form a compact group $\mathcal{G}(H)$ and then form an algebra of representative functions $\mathcal{R}(\mathcal{G}(H))$. The theorem states that:

$H$ is isomorphic to $\mathcal{R}(\mathcal{G}(H))$

The isomorphism is the same as in Gelfand–Najmark and some steps in the proof are similar. Using Stone–Weierstrass theorem one shows that the image (call it $V$) of the canonical "to be" isomorphism is dense in $\mathcal{R}(\mathcal{G}(H))$ and one needs to check that it is closed. Then the authors claim that it is enough to prove that $V$ is a $\mathcal{G}(H)$-module (meaning that there is an action, say from the left, of $\mathcal{G}(H)$ on $V$ such that $g \triangleright ( \cdot)$ is linear) and they refer to the book of Brocker and Dieck "Representations of compact Lie group". I've checked this theorem and have two questions:

• Q1. It is about compact Lie groups. In Gracia–Bondia's book there is nothing about the Lie group structure on $\mathcal{G}(H)$. How to fix this problem?

• Q2. This theorem provides equivalent conditions for a $G$-submodule of $\mathcal{R}(G)$ to be closed. I don't see why these conditions are (if really they are!) automatically satisfied in out situation. Is this really the case?

Further, in Gracia–Bondia's book the authors formulate a second theorem:

If $G$ is a compact Lie group then $\delta \colon x \mapsto \delta_x$ (where $\delta_x(f)=f(x)$) is an isomorphism between compact Lie groups $G$ and $\mathcal{G}(\mathcal{R}(G))$.

I've checked the proof and as far as I understood the steps the situations is the following:

a) the fact that it is an group homomorphism uses only algebraic structures
b) injectivity follows from Peter-Weyl thm. and continuity is immediate.

The problem is with surjectivity: let us denote $\mathcal{G}:=\mathcal{G}(\mathcal{R}(G))$ and $\hat{}:\mathcal{R}(G) \to \mathcal{R}(\mathcal{G})$ is an isomorphism from the previous theorem. One shows that $\delta^*:F \mapsto F \circ \delta$ is right inverse of $\hat{}$ and therefore is also an isomorphism. Since $\mathcal{R}(G)$ and $\mathcal{R}(\mathcal{G})$ are dense in $C(G)$ and $C(\mathcal{G})$ resp. it follows that $\delta^*$ extends to the isomomorhism between $C(G)$ and $C(\mathcal{G})$ and thus $\delta$ is surjective. A continuous bijection beetween compact spaces is a homeomorphism, and a continuous homomorphism of Lie groups is smooth so we are done. But here is my third question:

• Q3. Where in the proof we used the fact that $G$ is a Lie group? Do we really need this asumption?

The proof uses the previous theorem but in the formulation of the previous theorem there is nothing about Lie structure. It seems to me that this Lie structure is somehow hidden. So, to summarize let me ask:

• Q4. Which of this two theorems has to be reformulated (if any) and how should it look like? Are the proofs which I've sketched here correct?

And the final question:

• Q5. why it doesn't work for $\mathbb{C}$? Where is a key point of the argument?
• Tannaka-Krein duality is about compact topological groups. The differential structure plays no role. For a Hopf-algebraic proof (after which Gracia-Bondía's is probably patterned) of the Tannaka-Krein theorem, check G. Hochschild's "The Structure of Lie Groups" (Holden-Day, 1965). Feb 11 '14 at 1:35
• Thank You for Your comment. I saw some aproaches to Tannaka-Krein in general topological setting but it was formulated in the language of representations and category theory---nothing about Hopf algebra structure (for example in Hewitt and Ross). Unfortunately I haven't book of Hochschild (which was by the way cited in Gracia's book) and was unable to find it on the internet. Do You know any other book's in which this topic is discussed (with the Hopf algebraic aproach)? Feb 11 '14 at 15:29
• The proof in Hewitt-Ross is written in a rather old-fashioned way, but the "dual object" they consider, called a "Krein algebra", is just the same as the Hopf algebra considered by Hochschild and Gracia-Bondía et al.. Tannaka-Krein duality also holds for linear algebraic groups, and the proof, which also uses the Hopf algebra structure of the dual, is very similar (there is no analog of the normalized Haar integral, though - here algebraicity replaces continuity). You can find a part of it in the book of T. A. Springer, "Linear Algebraic Groups" (2nd. ed., Springer-Verlag, 1998), Sec. 2.5. Feb 11 '14 at 15:50

On Q1 and Q3: The Lie structure is not needed, and that's why you don't see it. Pedro Lauridsen Ribeiro also referred to this in the comments. You also said “a continuous homomorphism of Lie groups is smooth”, which implies that if two Lie groups are isomorphic as topological groups, then they are isomorphic as Lie groups.

On Q4: The theorems are correct, and don't need to be reformulated. Your sketches also seem to be correct.

On Q5: It does not work for $\mathbb{C}$, because that is not a compact Lie group. You used compactness, for example in saying that a “continuous bijection beetween compact spaces is a homeomorphism”.

[On a sidenote: this whole Tannaka–Krein duality business works in a lot of different settings. For a baby case, let me shamelessly plug my bachelor's thesis. It gives the proof for finite groups, and also computes what goes wrong in the case you take $\mathbb{Z}$. This might help you understand why compactness is needed.]

• If Lie structure is not needed and if theorems need not to be reformulated then certainly the first proof must be reformulated because it relies on the theorem from Brocker's book which is about Lie groups. So my question would be: how to modify the proof in order to work also with the general topological groups? For Q5 let me put my question in more detail: why we consider only homomorphism into $\mathbb{R}$ and why $H$ is assumed to be a Hopf algebra over $\mathbb{R}$ and not over $\mathbb{C}$? Feb 11 '14 at 15:16
• Tannaka-Krein duality is really very general. You can actually start with a commutative Hopf algebra over any field, it does not have to be $\mathbb{R}$ or $\mathbb{C}$ at all. (See my bachelor's thesis; where I deal with representations of finite groups over arbitrary fields.) The general theory is dealt with by e.g. Deligne-Milne "Tannakian categories" (jmilne.org/math/xnotes/tc.html) or Deligne "Catégories tannakiennes" (publications.ias.edu/deligne/paper/406).
– jmc
Feb 11 '14 at 18:46
• I would however start with the case of compact topological groups, and continuous representations. See for example: en.wikipedia.org/wiki/Tannaka-Krein_duality which explains some ideas.
– jmc
Feb 11 '14 at 18:48