*-Structures on Module-Algebras

This chapter lays out a framework for discussing (\ast)-structures on module-algebras over a Hopf (\ast)-algebra (H). We define a complex conjugation functor (V \mapsto \bar{V}), which is an involution on the module category (\hmod), and discuss its interaction with natural constructions such as direct sums, duality, Hom, and tensor products. We define (\ast)-structures first at the level of modules. We say that (V) is a (\ast)-module if there is an isomorphism (\ast : \bar{V} \to V) in (\hmod) which is involutive in an appropriate sense. Then we define (\ast)-structures on algebras in (\hmod) by requiring compatibility with multiplication. We show that a (\ast)-structure on a module lifts uniquely to the tensor algebra, and we prove that the tensor algebra has a universal mapping properly for morphisms of (\ast)-modules. We also discuss inner products and adjoints in this framework. Finally, we discuss the interaction between (\ast)-structures, (R)-matrices, and braidings.

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