Transmutation and Bosonisation of Quasi-Hopf Algebras

Let $H$ be a quasitriangular quasi-Hopf algebra, we construct a braided group $\underset{‾}{H}$ in the quasiassociative category of left $H$-modules. Conversely, given any braided group $B$ in this category, we construct a quasi-Hopf algebra $B⋊H$ in the category of vector spaces. We generalise the transmutation and bosonisation theory of [10] to the quasi case. As examples, we bosonise the octonion algebra to an asoociative one, obtain the twisted quantum double $D^{\varphi }(G)$ of a finite group as a bosonisation, and obtain its transmutation. Finally, we show that $\underset{‾}{H}⋊H$ is isomorphic to $H_{\text{Rcal}}\text{blackbowtie}H$ as quasi-Hopf algebras.