2 00 6 Bar categories and star operations

We introduce the notion of 'bar category' by which we mean a monoidal category equipped with additional structure induced by complex conjugation. Examples of our theory include bimodules over a *-algebra, modules over a conventional Hopf *-algebra and modules over a more general object which call a 'quasi-*-Hopf algebra' and for which examples include the standard quantum groups u q (g) at q a root of unity (these are well-known not to be a usual Hopf *-algebra). We also provide examples of strictly quasiassociative bar categories, including modules over ' *-quasiHopf algebras' and a construction based on finite subgroups H ⊂ G of a finite group. Inside a bar category one has natural notions of '⋆-algebra' and 'unitary object' therefore extending these concepts to a variety of new situations. We study braidings and duals in bar categories and ⋆-braided groups (Hopf algebras) in braided-bar categories. Examples include the transmutation B(H) of a quasitriangular Hopf *-algebra and the quantum plane C 2 q at certain roots of unity q in the bar category u q (su 2)-modules. We use our methods to provide a natural quasi-associative C *-algebra structure on the octonions O and on a coset example. In the appendix we extend the Tannaka-Krein reconstruction theory to bar categories in relation to Hopf *-algebras.