The Half-quantum Group U ≥0

Let U denote the half quantum group for a fixed simple Lie algebra. We examine some properties and representation of U. We prove that the Hopf algebra U is not quasi-cocommutative, and hence the category of left U-module is not a braided monoidal category. In the weight module category W , we describe all the simple objects and the projective objects. We also describe all simple Yetter-Drinfel’d U-weight modules.