The duality theory of a finite dimensional discrete quantum group

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