超多元结构
Hyperpolyadic structures
AI总结：
本文引入了对应于二元除法代数的超多元代数，通过矩阵多元过程和“虚塔”迭代构造了不同维数的 n 元除法代数，并提出了“半四元数”等新结构及其相关范数恒等式。
摘要：
我们引入了一类新的除法代数，即超多元代数（hyperpolyadic algebras），它们对应于二元除法代数 $\mathbb{R}$、$\mathbb{C}$、$\mathbb{H}$、$\mathbb{O}$，而无需考虑新的元素。首先，我们使用了先前提出的矩阵多元过程，该过程增加了代数的维数。以这种方式获得的代数遵循二元加法以及非导出的 n 元乘法，且其子代数为 n 元除法代数。对于每个可逆元素，我们定义了一个新的范数，该范数是多元乘法的，相应的映射是一个 $n$ 元同态。我们定义了 Cayley-Dickson 构造的多元类似物，它对应于多元过程中单项矩阵的后续嵌入。然后，我们获得了对应于二元除法代数的另一系列 n 元代数，它们具有更高的维数，且该维数与中间元数成正比。其次，定义了任意向量空间中向量的新多元乘积。赋予此乘积后，向量空间成为一个多元代数，该代数在某些可逆性条件下是除法代数，并且计算了其结构常数。第三，我们提出了一种新的迭代过程（“虚塔”），它导致了维数减半的非幺元非导出三元除法代数，我们称之为“半四元数”和“半八元数”。后者不是二元除法代数的子代数，而仅仅是子集，因为它们具有不同的元数。然而，它们实际上是三元除法代数，因为它们允许除法，且其非零元素是可逆的。由所引入的“半四元数”范数的乘法性，我们得到了两平方和恒等式的三元类似物。我们证明了虚“半八元数”的三元除法代数是无单位的且完全结合的。
英文摘要：
We introduce a new class of division algebras, the hyperpolyadic algebras, which correspond to the binary division algebras $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, $\mathbb{O}$ without considering new elements. First, we use the matrix polyadization procedure proposed earlier which increases the dimension of the algebra. The algebras obtained in this way obey binary addition and a nonderived n-ary multiplication and their subalgebras are division n-ary algebras. For each invertible element we define a new norm which is polyadically multiplicative, and the corresponding map is a $n$-ary homomorphism. We define a polyadic analog of the Cayley-Dickson construction which corresponds to the consequent embedding of monomial matrices from the polyadization procedure. We then obtain another series of n-ary algebras corresponding to the binary division algebras which have a higher dimension, that is proportional to the intermediate arities. Second, a new polyadic product of vectors in any vector space is defined. Endowed with this product the vector space becomes a polyadic algebra which is a division algebra under some invertibility conditions, and its structure constants are computed. Third, we propose a new iterative process ("imaginary tower"), which leads to nonunital nonderived ternary division algebras of half the dimension, which we call "half-quaternions" and "half-octonions". The latter are not subalgebras of the binary division algebras, but subsets only, since they have different arity. Nevertheless, they are actually ternary division algebras, because they allow division, and their nonzero elements are invertible. From the multiplicativity of the introduced "half-quaternion" norm we obtain the ternary analog of the sum of two squares identity. We show that the ternary division algebra of imaginary "half-octonions" is unitless and totally associative.
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