Co-Moufang deformations of the universal enveloping algebra of the algebra of traceless octonions

Cayley平面的一个极简且非交替的实现

A minimal and non-alternative realisation of the Cayley plane

本研究用更弱的非交替代数（para八元数和Okubo代数）简洁地定义了Cayley平面，并证明实数Okubo代数是最弱可定义紧致16维Moufang平面的代数结构。

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 $G$ 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 $S:H\to H$, 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…

A series of algebras generalizing the octonions and Hurwitz-Radon identity

We study non-associative twisted group algebras over $(\Z_2)^n$ with cubic twisting functions. We construct a series of algebras that extend the classical algebra of octonions in the same way as the Clifford algebras extend the algebra of quaternions. We study their properties, give several equivalent definitions and prove their uniqueness within some natural assumptions. We then prove a simplicity criterion. We present two applications of the constructed algebras and the developed technique. The first application is a simple explicit formula for the following famous square identity: $(a_1^2+.…