Bar categories and star operations

Abstract

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 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 $H\subset G$ of a finite group. Inside a bar category one has natural notions of `$\star$-algebra' and `unitary object' therefore extending these concepts to a variety of new situations. We study braidings and duals in bar categories and $\star$-braided groups (Hopf algebras) {\em in} braided-bar categories. Examples include the transmutation $B(H)$ of a quasitriangular $*$-Hopf algebra and the quantum plane $C_q^2$ 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.