2009/06/27 by J. Klim, Shahn Majid, Klim, J. +2 · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Mathematics and Applications #Quantum Algebra (math.QA) #Rings and Algebras (math.RA) #math.QA #math.RA
paper · pdf · doi:10.48550/arxiv.0906.5026
43 pages latex; added Maurer-Cartan equation (Prop 6.5) and computation of it for S^7 (lemma 6.8). No other change aside typos
openalex publication_date 2009/06/27 · arxiv created 2009/12/15 · arxiv updated 2010/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce the notions of Hopf quasigroup and Hopf coquasigroup H generalising the classical notion of an inverse property quasigroup G expressed respectively as a quasigroup algebra k G and an algebraic quasigroup k[G]. We prove basic results as for Hopf algebras, such as anti(co)multiplicativity of the antipode S:H→ H, that S2=\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[S2n-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 \citeMa99. We construct an example k[S7]\rtimes\Z23 of a Hopf coquasigroup which is noncommutative and non-trivially Moufang. We use Hopf coquasigroup methods to study differential geometry on k[S7] including a short algebraic proof that S7 is parallelizable. Looking at combinations of left and right invariant vector fields on k[S7] we provide a new description of the structure constants of the Lie algebra g2 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 \Cq[S3] 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.