Transmutation and Bosonisation of Quasi-Hopf Algebras

TL;DR

This paper extends the transmutation and bosonisation theories to quasi-Hopf algebras, constructing braided groups and new quasi-Hopf algebra structures with applications to octonions and quantum doubles.

Contribution

It generalizes the transmutation and bosonisation framework to quasi-Hopf algebras, providing new constructions and examples including octonions and twisted quantum doubles.

Findings

Bosonisation of octonion algebra yields an associative algebra.

Construction of twisted quantum double $D^{}(G)$ as a bosonisation.

Isomorphism between $al{H} times H$ and $H_ cal lackbowtie H$ as quasi-Hopf algebras.

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 a quasi-Hopf algebra $B\rtimes H$ 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 $D^{\phi}(G)$ of a finite group as a bosonisation, and obtain its transmutation. Finally, we show that $\underline{H}\rtimes H$ is isomorphic to $H_\Rcal\blackbowtie H$ as quasi-Hopf algebras.