Bar Categories and Star Operations

We introduce the notion of ‘bar category’ by which we mean a monoidal category equipped with additional structure formalising the notion of 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 we call a ‘quasi-*-Hopf algebra’ and for which examples include the standard quantum groups $u_q(\mathfrak{g})$ at q a root of unity (these are well-known not to be usual *-Hopf algebras). 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 ${\mathbb C}_q^2$ at certain roots of unity q in the bar category of $\widetilde{u_q(su_2)}$ -modules. We use our methods to provide a natural quasi-associative C ^*-algebra structure on the octonions ${\mathbb O}$ and on a coset example. In the Appendix we extend the Tannaka-Krein reconstruction theory to bar categories in relation to *-Hopf algebras.