An Introduction to Tannaka Duality and Quantum Groups

The goal of this paper is to give an account of classical Tannaka duality [C⁄] in such a way as to be accessible to the general mathematical reader, and to provide a key for entry to more recent developments [⁄SR, DM⁄] and quantum groups [⁄D1⁄]. Expertise in neither representation theory nor category theory is assumed. Naively speaking, Tannaka duality theory is the study of the interplay which exists between a group and the category of its representations. The early duality theorems of Tannaka-Krein [Ta, Kr] concentrate on the problem of reconstructing a compact group from the collection of its representations. In the abelian case, this problem amounts to reconstructing the group from its character group, and is the content of the Pontrjagin duality theorem. A good exposition of this theory can be found in the book by Chevalley [C]. In these early developments, there was little or no use of categorical concepts, partly because they did not exist at the time. Moreover, the mathematical community was not yet familiar with category theory, and it was possible to avoid it [BtD]. To Grothendieck we owe the understanding that the process of Tannaka duality can be reversed. In his work to solve the Weil conjectures, he constructed the category of motives as the universal recipient of a Weil cohomology [Kl]. By using a fiber functor from his category of motives to vector spaces, he could construct a pro-algebraic group G. He also conjectured that the category of motives could be recaptured as the category of representations of G. This group is called the Grothendieck Galois group, since it is an extension of the Galois group of _Q⁄⁄/Q . The work spreading from these ideas can be found in [SR, DM]. For other aspects of this question, see [Cb]. An entirely different development came from mathematical physicists working on superselection principles in quantum field theory [DHR] where it was discovered that the superselection structure could be described in terms of a category whose objects are certain endomorphisms of the C*-algebra of local observables, and whose arrows are intertwining operators. Reversing the duality process, they succeeded in constructing a compact group whose representations can be identified with their superselection category [DR]. Another impulse to the development of Tannaka duality comes from the theory of quantum groups. These new mathematical objects were discovered by Jimbo [⁄J⁄] and Drinfel'd [D1] in connection with the work of L.D. Faddeev and his collaborators on the quantum inverse scattering method. V.V. Lyubashenko [Ly] initiated the use of Tannaka duality in the construction of quantum groups; also see K.-H. Ulbrich [U]. We should also mention S.L. Woronowicz [W] in the case of compact quantum groups. Recently, S. Majid [M3] has shown that one can use Tannaka-Krein duality for constructing the quasi-Hopf algebras introduced by Drinfel'd [D2] in connection with the solution of the KnizhnikZamolodchikov equation. The theory of angular momentum in Quantum Physics [BL1] might also provide some extra motivation for studying Tannaka duality. The Racah-Wigner algebra, the 9⁄⁄– ⁄⁄j and 3⁄⁄– ⁄⁄j symbols, and, the Racah and Wigner coefficients, all seem to be about the explicit description of the structures which exist on the category of representations of some