The duality theory of a finite dimensional discrete quantum group

Abstract

Abstract. 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.As an application, a new proof of the classical Schur-Weyl duality theory of type A is given.

On Quantum Duality of Group Amenability

“…In the article, all algebras are considered over the complex field C. For the details on CQGs and C*-norms, we refer to [6][7][8][9][10][11][12][13]; and for the general conclusions for pairing and quantum double, we refer to [2,6,[14][15][16][17]. In our proofs, we make use of a large quantity of calculations by the standard Sweedler notation.…”