nLab
Doplicher-Roberts reconstruction theorem
Contents
Context
Representation theory
representation theory

geometric representation theory

Ingredients
Definitions
representation , 2-representation , ∞-representation

group , ∞-group

group algebra , algebraic group , Lie algebra

vector space , n-vector space

affine space , symplectic vector space

action , ∞-action

module , equivariant object

bimodule , Morita equivalence

induced representation , Frobenius reciprocity

Hilbert space , Banach space , Fourier transform , functional analysis

orbit , coadjoint orbit , Killing form

unitary representation

geometric quantization , coherent state

socle , quiver

module algebra , comodule algebra , Hopf action , measuring

Geometric representation theory
D-module , perverse sheaf ,

Grothendieck group , lambda-ring , symmetric function , formal group

principal bundle , torsor , vector bundle , Atiyah Lie algebroid

geometric function theory , groupoidification

Eilenberg-Moore category , algebra over an operad , actegory , crossed module

reconstruction theorems

Theorems
Contents
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 conveived 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 ).

References
Original statement and proof in the context of AQFT :

Generalization to supergroups and streamlined proof:

See also:

Last revised on March 24, 2021 at 00:26:56.
See the history of this page for a list of all contributions to it.