New approach to octonions and cayley algebras

Quasialgebra structure of the octonions

We show that the octonions are a twisting of the group algebra of Z_2 x Z_2 x Z_2 in the quasitensor category of representations of a quasi-Hopf algebra associated to a group 3-cocycle. We consider general quasi-associative algebras of this type and some general constructions for them, including quasi-linear algebra and representation theory, and an automorphism quasi-Hopf algebra. Other examples include the higher 2^n-onion Cayley algebras and examples associated to Hadamard matrices.

The octonions form an Azumaya algebra in certain braided linear Gr-categories

By applying the idea of viewing the octonions as an associative algebra in certain tensor categories, or more precisely as a twisted group algebra by a 2-cochain, we show that the octonions form an Azumaya algebra in some suitable braided linear Gr-categories.

A quantum octonion algebra

Using the natural irreducible 8-dimensional representation and the two spin representations of the quantum group $U_q$(D$_4$) of D$_4$, we construct a quantum analogue of the split octonions and study its properties. We prove that the quantum octonion algebra satisfies the q-Principle of Local Triality and has a nondegenerate bilinear form which satisfies a q-version of the composition property. By its construction, the quantum octonion algebra is a nonassociative algebra with a Yang-Baxter operator action coming from the R-matrix of $U_q$(D$_4$). The product in the quantum octonions is a $U_q…

On the conjugacy of pure imaginary elements of quaternion algebras and Cayley algebras

In the quatenions ($\H$, $\H'$, $\H^{C}$) and octonions ($\gC$, $\gC^\prime$, $\gC^C$), we show some results on the conjugacy of two pure imaginary non-zero elements with same norm.

The Exponential Map on the Cayley-Dickson algebras

we study the exponential map for A_n = R^2^n, the Cayley_Dickson algebras for n bigher than 1,wich generalize the Complex exponential map to Quaternions,Octonions and so forth. As application,we show that the self-map of the unit sphere in A_n, S^(2^n)-1,given by taking k-powers has topological degree k for k an integer number,from this we derive a suitable "Fundamental theorem of algebra" for A_n for n bigher than 1.

通过张量范畴从八元数到复合超代数

From octonions to composition superalgebras via tensor categories

本文通过特征3循环群表示的对称张量范畴的半简化过程，从分裂Cayley代数及其3阶自同构获得仅存在于特征3的3维和6维非平凡含幺复合超代数，并讨论其与扩展Freudenthal魔方及例外Lie超代数的联系。