Hopf quasigroups and the algebraic 7-sphere
We introduce the notions of Hopf quasigroup and Hopf coquasigroup H generalising the classical notion of an inverse property quasigroup inlMMLBox expressed respectively as a quasigroup algebra inlMMLBox and an algebraic quasigroup inlMMLBox . We prove basic results as for Hopf algebras, such as anti(co)multiplicativity of the antipode inlMMLBox , that S 2 =id if H is commutative or cocommutative, and a theory of crossed (co)products. We also introduce the notion of a Moufang Hopf (co)quasigroup and show that the coordinate algebras k [ S 2 n −1 ] of the parallelizable spheres are algebraic quasigroups (commutative Hopf coquasigroups in our formulation) and Moufang. We make use of the description of composition algebras such as the octonions via a cochain F introduced in  . We construct an example inlMMLBox of a Hopf coquasigroup which is noncommutative and nontrivially Moufang. We use Hopf coquasigroup methods to study differential geometry on k [ S 7 ] including a short algebraic proof that S 7 is parallelizable. Looking at combinations of left- and right-invariant vector fields on k [ S 7 ] we provide a new description of the structure constants of the Lie algebra g 2 in terms of the structure constants F of the octonions. In the concluding section we give a new description of the q -deformation quantum group inlMMLBox regarded trivially as a Moufang Hopf coquasigroup (trivially since it is in fact a Hopf algebra) but now in terms of F built up via the Cayley–Dickson process.