Talk:Tannaka–Krein duality
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[edit]" Its natural extension to the non-Abelian case is the Grothendieck duality theory. "
I think this statement is incorrect. Tannaka-Krein duality is the natural extension to the non-Abelian case of the Pontryagin duality. I don't know about Grothendieck duality.
Thanks Charles(I meant Charles, spaced today), whoever you are:) I started this because I saw it had been on the requested list for two years. My mathematical interest is very much a lay-persons. I figured someone with a far better grasp of mathematics than I could expand based on their own knowledge, an understanding of the links, or both.--T. Anthony 01:29, 3 September 2005 (UTC)
one cannot reconstruct a finite group from its character table
[edit]One cannot reconstruct a finite group from its character table, e.g. have the same character table. But one can reconstruct them from their representation category by Tannaka duality. Can someone explain what the additional information contained in the representation category is? Thanks. Ringspectrum (talk) 22:43, 17 January 2009 (UTC)
- --Gherardo (talk) 07:30, 18 March 2011 (UTC) The objects in the Tannaka-Krein category are vector spaces. The objects in the Doplicher Roberts category are abstract, only the categorical structure is relevant. An example of application from mathematical physics (which was the motivation of Doplicher and Roberts). In quantum field theory, the observable algebra is described, together with all its charged sectors (=superselection structure), as the fix point subalgebra of the (operator) field algebra under the action of a group of unitary operators (global gauge group). Multiplets of unobservable fields realise the Tannaka-Krein category, and are associated with the morphisms of the observable algebra. In principle, the observable algebra in the vacuum sector should contain all the informations about the theory. The problem then was: is it possible to reconstruct the gauge group, the unobservable fields and the superselection structure, starting from the observable algebra in the vacuum sector? Here the would-be dual category has the morphisms of the observable algebra as it objects, there is no way to understand them as vector spaces, and the Tannaka Krein duality cannot be used. The DR duality can.