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 θ 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.