Tannaka–Krein duality

Remarks

This is Tannaka's theorem: a compact group is determined by its category of representations together with the forgetful functor. Proof idea: by the Peter–Weyl theorem, the matrix coefficients of the representations span a dense subalgebra $\mathcal{R}(G)$ of the algebra $C(G)$ of continuous functions. These functions also separate the points of $G$. An element of $T(G)$ defines an algebra homomorphism $\mathcal{R}(G) \to \C$ that commutes with complex conjugation. One shows that every such homomorphism is evaluation at a unique point of $G$.

Without clause 3 the group is larger. For $G = U(1)$, the representations are the $\Z$-graded spaces, and families satisfying clauses 1 and 2 are given by $t \in \C^\times$ acting by $t^n$ in degree $n$; clause 3 forces $\abs{t} = 1$. For a compact Lie group, clauses 1 and 2 alone give its complexification. Krein's part of the duality describes which categories with such a functor arise from compact groups. The algebraic version concerns a neutral Tannakian category over a field $k$. It says that such a category is equivalent, as a rigid symmetric monoidal category, to the category of finite-dimensional representations of an affine group scheme over $k$.

Going deeper

For a student

For a compact abelian group, every irreducible representation is one-dimensional, and the theorem reduces to recovering $G$ from its character group, as in Pontryagin duality. For non-abelian groups the representations do not form a group, but they do form a rigid symmetric monoidal category, and the forgetful functor is a strong monoidal functor. The group appears as the automorphisms of this functor that respect the tensor product. This idea is central to the theory of Tannakian categories, motivic Galois groups, and the reconstruction of quantum groups from their representation categories.