# Transmutation and Bosonisation of Quasi-Hopf Algebras - Mathematics > Quantum Algebra

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Abstract: 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 $B times H$ 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 $\underline{H} times H$ isisomorphic to $H \Rcal\blackbowtie H$ as quasi-Hopf algebras.

Author: ** J Klim**

Source: https://arxiv.org/