Homomorphisms of quantum groups (Q2869223)

The paper uses the definition of a locally compact group based on modular multiplicative unitaries due to \textit{P. M. Sołtan} and \textit{S. L. Woronowicz} [J. Funct. Anal. 252, No. 1, 42--67 (2007; Zbl 1134.46044)]. The approach to the homomorphisms developed is based on the point of view that quantum groups encode symmetries of \(C^*\)-algebras, in the form of coactions. More concretely, let \(\mathfrak{C^*alg}(A)\) be the category of \(C^*\)-algebras with a continuous coaction of a \(C^*\)- bialgebra \((A,\Delta_A)\), with \(A\)-equivariant morphisms as arrows. Forgetting the coaction provides a functor \(\mathrm{\mathbf For}\) to the category \(\mathfrak{C^*alg}\) of \(C^*\)-algebras without extra structure. Then one defines a quantum group homomorphism from a locally compact quantum group \((C,\Delta_C)\) to \((A,\Delta_A)\) is a functor \(F:\mathfrak{C^*alg}(C)\to\mathfrak{C^*alg}(A)\) with \(F\circ\mathrm{\mathbf For}=\mathrm{\mathbf For}\).NEWLINENEWLINENg introduced a notion of quantum group morphism in [\textit{C.-K. Ng}, J. Oper. Theory 38, No. 2, 203--224 (1997; Zbl 0907.46049)]. The results in this paper show that Ng's theory works, provided that the quantum groups are defined by multiplicative unitaries that lift to the universal quantum groups.