The duality theory of a finite dimensional discrete quantum group

Suppose that H is a finite dimensional discrete quantum group and K is a Hilbert space. This paper shows that if there exists an action γ of H on L(K) so that L(K) is a modular algebra and the inner product on K is H-invariant, then there is a unique C*-representation 9 of H on K supplemented by the γ. The commutant of θ(H) in L(K) is exactly the H-invariant subalgebra of L(K). As an application, a new proof of the classical Schur-Weyl duality theory of type A is given.