Tannaka-Krein duality
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In mathematics, Tannaka-Krein duality theory concerns the interaction of a topological group and its category of linear representations. The aim is for a version of Pontryagin duality that is a mathematical duality for groups that are not commutative. The case of compact groups was first treated successfully. The theory is named for two men, the Ukrainian mathematician Mark Grigorievich Krein, and the Japanese Tannaka.
The theorem is relevant to quantum groups and general mathematical physics. Alexander Grothendieck showed that the process of Tannaka duality can be extended to algebraic groups: see tannakian category.
[edit] Doplicher-Roberts theorem
This result[1] characterises Rep(G) in terms of category theory, as a type of subcategory of the category of Hilbert spaces. Such categories of unitary representation of a compact group are the same as certain subcategories, the required properties being:
- a strict symmetric monoidal C*-category with conjugates
- having subobjects and direct sums, such that
- the C*-algebra of endomorphisms of the monoidal unit is of scalars.
[edit] Notes
- ^ S. Doplicher and J. Roberts. A new duality theory for compact groups. Inventiones Mathematicae, 98:157--218, 1989.
