Archive for July 2025

The Quill 20 ~ Tannaka-Krein Duality and Reconstruction

Tannaka-Krein duality, in its classical form, says that given a compact group $G$, one can reconstruct the group $G$ from the category of the finite-dimensional complex representations $\Pi(G)$. And $\Pi(G)$ is a tensor category (thus having a monoidal structure). For a locally compact abelian group, it is a much easier case and from Pontryagin duality, one has that the dual group $\hat{G}$ is the space of 1-dimensional representations of G (namely characters), and the dual group determines the group $G$, please see this note from last year. However, for non-abelian cases, the category of representation $\Pi(G)$ contains all finite-dimensional $\mathbb{C}-$linear representations, not just 1-dimensional irreducible representations like abelian case. Actually, as I think, it is helpful to see that Pontryagin duality is an abelian prelude to Tannaka-Krein duality. The latter also generalized a lot of the framework later, which culminated in the Tannakian framework of Grothendieck, quantum groups by Drinfeld, Deligne's tensor categories, and so on.

But one can say that given
$$\Pi(G) \cong \Pi(H)$$
which is an equivalence of symmetric monoidal categories, $G \cong H$ will not always be true unless there are some conditions to observe. This was emphasized by introducing a faithful, exact, forgetful, $\mathbb{C}$-linear fiber functor which preserves the tensor products and forgets the group action $\omega$
$$\omega: \Pi(G) \rightarrow Vect_\mathbb{C}$$
which forgets the group action and remembers the underlying vector spaces. The group $G$ can now be associated with the automorphism of the fiber functor
$$G \cong \underline{Aut}^{\otimes}(\omega).$$
This essentially means that all the information about the group is encoded in $\Pi(G)$, and given this fiber functor, we can reconstruct our group. In more standard words, given any symmetrical monoidal category $\mathcal{C}$, if one has a fiber functor $\omega: \mathcal{C} \rightarrow Vect_\mathbb{C}$, then under the suitable conditions, there exists a compact group $G$ such that $\mathcal{C} \cong \Pi(G)$ and $G  \cong \underline{Aut}^{\otimes}(\omega)$.

The framework of Tannakian categories and fiber functor was introduced by Grothendieck-Deligne in the 1960s, while Tannaka originally used the ring of representative functions.

Posted in | Leave a comment Print it.