推广八元数的一系列代数与Hurwitz-Radon恒等式
A series of algebras generalizing the octonions and Hurwitz-Radon identity
摘要：
研究了$(\Z_2)^n$上具有三次扭函数的非结合扭群代数.我们构造了一系列的代数，以同样的方式扩展的经典代数的八元数的Clifford代数扩展的四元数。我们研究了它们的性质，给出了几个等价定义，并在一些自然假设下证明了它们的唯一性。然后，我们证明了一个简单的标准。我们提出了两个应用所构造的代数和开发的技术。第一个应用是一个简单的显式公式为以下著名的平方恒等式：$(a_1^2+...+a_{N}^2)\,(b_1^2+...+b_{ρ(N)}^2)= c_1^2+...+c_{N}^2$，其中$c_k$是双线性函数的$a_i$和$b_j$和其中$ρ(n)$是Hurwitz-Radon函数。第二个应用是与Moufang循环的关系，特别是与代码循环的关系。为了说明这种关系，我们提供了一个明确的坐标公式的因子组的帕克循环。
英文摘要：
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: $(a_1^2+...+a_{N}^2)\,(b_1^2+...+b_{ρ(N)}^2)= c_1^2+...+c_{N}^2$, where $c_k$ are bilinear functions of the $a_i$ and $b_j$ and where $ρ(n)$ is the Hurwitz-Radon function. The second application is the relation to Moufang loops and, in particular, to the code loops. To illustrate this relation, we provide an explicit coordinate formula for the factor set of the Parker loop.
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