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An introduction to Tannaka duality and quantum groups. (English) Zbl 0745.57001

Category theory, Proc. Int. Conf., Como/Italy 1990, Lect. Notes Math. 1488, 413-492 (1991).
[For the entire collection see Zbl 0733.00009.]
This paper aims at introducing the general mathematical reader to some modern developments which indicate the emergence of a new symbiosis between algebra, geometry and physics. A categorical approach to Tannaka duality is taken as the unifying theme.
The early duality theorems of Tannaka-Krein concentrated on the problem of reconstructing a compact group from the collection of its representations. Grothendieck reversed the process (in his work on the Weil conjectures) by constructing a pro-algebraic group from his category of motives. Also, mathematical physicists working on superselection principles in quantum field theory discovered that the superselection structure could be described in terms of a category whose objects were certain endomorphisms of the \(C^*\)-algebra of local observables, and, by reversing the duality process, they could construct a compact group whose representations could be identified with their superselection category. After the discovery by M. Jimbo, V. G. Drinfel’d and S. L. Woronowicz of new objects, called quantum groups, V. V. Lyubashenko showed how to use Tannaka duality in their construction. Other related areas where Tannaka duality is of interest are the theory of angular momentum in quantum physics, and the construction of knot invariants.
After a brief review of Pontryagin duality and Fourier transform for locally compact abelian groups, the paper gives a treatment of the classical Tannaka theory for compact (non-abelian) groups. A central object of the analysis is the algebra \(R(M)\) of representative complex- valued functions on a topological monoid \(M\). Then comes the modern treatment of Tannaka reconstruction, motivated at each stage by the example of a monoid \(M\). Instrumental here is the Fourier cotransform which can be seen as the continuous predual of the Fourier transform. In fact, the Fourier cotransform provides an isomorphism between the reconstructed object and \(R(M)\). An independent introduction to Tannaka duality for homogeneous spaces is included. Categories of comodules over a coalgebra are characterized. Hopf algebras, braided tensor categories and Yang-Baxter operators are discussed, with examples. A brief description of the categorical axiomatization of the geometry of knots and tangles is provided. Finally there is a too brief introduction to quantum groups.
Reviewer: R.Street

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
43A95 Categorical methods for abstract harmonic analysis
22A25 Representations of general topological groups and semigroups
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)

Citations:

Zbl 0733.00009
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