Quasi-Hopf algebras and representations of octonions and other quasialgebras

Modules over a quasialgebra (here, by quasialgebra we mean a left $H$-module algebra, where $H$ is a quasi-Hopf algebra), as defined by Albuquerque and Majid, coincide with modules over a certain associative algebra, a quasi-Hopf smash product. As a consequence of this, we get that the category of modules over the octonions is isomorphic to the category of modules over the algebra of $8×8$ real matrices. We provide a new approach to the endomorphism quasialgebra associated to a left $H$-module, which in the finite dimensional case yields the same results as the one of Albuquerque and Majid. We discuss possible definitions as endomorphism quasialgebras for Heisenberg doubles of a finite dimensional quasi-Hopf algebra.