Isomorphic representations and groups

Question

Suppose $G_{1}, G_{2}$ are two groups. Suppose further that the sets of representations of both groups are isomorphic. When $G_{1}, G_{2}$ are compact topological groups, the Tannaka-Krein duality tells us that this implies that the groups themselves are isomorphic. Does a similar statement (both of Tannaka-Krein and the isomorphism of the groups) hold for non-compact groups?

Answer 1

For an abstract group $G$ (so with no topology), we have the following version of Tannaka-Krein:

Let $k$ be a field. Consider the category $\mathbf{Rep}_k(G)$ of representations of $G$ over $k$ and the category $k\textrm{-}\mathbf{Vect}$ of rerpresentations. Both are monoidal categories and the forgetful ("fibre") functor $F:\mathbf{Rep}_k(G) \to k\textrm{-}\mathbf{Vect}$ is a (strictly) monoidal functor. From this data we can first reconstruct $k[G]$ as a bialgebra as $\mathrm{End}(F)$ with the monoidal structure giving rise to the comultiplication. Then we can reconstruct $G$ as the grouplike elements in $k[G]$.