nLab Doplicher-Roberts reconstruction theorem

Idea

The Doplicher-Roberts reconstruction theorem allows one to reconstruct a compact topological group or “supergroup” (here essentially: $\mathbb{Z}/2$-graded group) from its category of representations, given enough extra structure on this category. Unlike some other versions of Tannaka duality, such as Deligne's theorem on tensor categories, it does not require knowledge of the forgetful functor (“fiber functor”) from the category of representations to VectorSpaces.

This result is named after Sergio Doplicher and John Roberts, who conceived of it in the context of algebraic quantum field theory as an intrinsic characterization of the superselection sectors of a quantum field theory. See at DHR superselection theory for more on this aspect. A simplified and self-contained proof was given in Müger 06.

In brief, the DR reconstruction theorem says that every symmetric tensor star-category with conjugates, direct sums, subobjects and endomorphism ring of the tensor unit isomorphic to the complex numbers, $End(1)= \mathbb{C}$, is equivalent to the category of finite dimensional unitary linear representations of a compact supergroup, which is unique up to isomorphism. For definitions of these terms, see Müger 06.

Beware that there are non-isomorphic finite groups whose categories of representations are equivalent considered merely as tensor categories, ignoring the symmetric structure (e.g. Etingof-Gelaki 00).

Related concepts

• Deligne's theorem on tensor categories

• Wigner classification

  • unitary representation of the Poincaré group

  • unitary representation of the super Poincaré group

References

Original statement and proof in the context of AQFT:

• Sergio Doplicher, John Roberts, A new duality theory for compact groups, Inventiones mathematicae 98.1 (1989): 157-218. (available here)

Generalization to supergroups and streamlined proof:

• Michael Müger, Abstract duality theory for symmetric tensor $\ast$-categories appendix (pdf), in Hans Halvorson, Algebraic quantum field theory (arXiv:math-ph/0602036), in J. Butterfield & J. Earman (eds.) Handbook of Philosophy and Physics

Review:

• Garth Warner: Reconstruction Theory, EPrint Collection, University of Washington (2011) [hdl:1773/16351, pdf, pdf]

See also:

• Pavel Etingof, Shlomo Gelaki, Isocategorical groups [arXiv:math/0007196]