The Tannaka–Krein duality provides a way to reconstruct a group (up to isomorphisms) from the category of linear representations of that group.
In physics, this duality is sometimes used as a justification for considering only linear representations in some context (e.g. when classifying elementary particles, see this question).
However, the duality as explained in Wikipedia is for compact groups and groups considered in physics are not compact (e.g. Poincaré group and its cover), only locally-compact.
What does this duality say in the non-compact case?