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Let $H$ be a quasitriangular quasi-Hopf algebra, we construct a braided group$\underline{H}$ in the quasiassociative category of left $H$-modules.Conversely, given any braided group $B$ in this category, we construct aquasi-Hopf algebra $B\rtimes H$ in the category of vector spaces. We generalisethe transmutation and bosonisation theory of [10] to the quasi case. Asexamples, we bosonise the octonion algebra to an asoociative one, obtain thetwisted quantum double $D^{\phi}(G)$ of a finite group as a bosonisation, andobtain its transmutation. Finally, we show that $\underline{H}\rtimes H$ isisomorphic to $H_\Rcal\blackbowtie H$ as quasi-Hopf algebras.

Transmutation and Bosonisation of Quasi-Hopf Algebras