$\Z_2^n$-graded quasialgebras and the Hurwitz problem on compositions of quadratic forms
$\Z_2^n$-graded quasialgebras and the Hurwitz problem on compositions of quadratic forms
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DOI：
10.1090/tran/6946
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发表时间：
2015-06
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影响因子：
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通讯作者：
Ya-Qing Hu;Hua-Lin Huang;Chi Zhang
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文献类型：
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作者：
Ya-Qing Hu;Hua-Lin Huang;Chi Zhang
引入了一系列推广了Clifford代数、高八元数和高Cayley代数的$\Z_2^n$阶拟代数$\bbP_n(m)$。利用所构造的代数级数及其微扰给出了二次型复合的Hurwitz问题的显式解。特别地，我们统一地给出了著名的Hurwitz-Radon平方恒等式的显式表达式，恢复了Yuzvinsky- lam - smith公式，确认了Yuzvinsky在1984年提出的第三类可允许三元组，改进了Lenzhen、Morier-Genoud和Ovsienko最近得到的两个无穷族解，构造了几个新的无穷族解。
We introduce a series of $\Z_2^n$-graded quasialgebras $\bbP_n(m)$ which generalizes Clifford algebras, higher octonions, and higher Cayley algebras. The constructed series of algebras and their minor perturbations are applied to contribute explicit solutions to the Hurwitz problem on compositions of quadratic forms. In particular, we provide explicit expressions of the well-known Hurwitz-Radon square identities in a uniform way, recover the Yuzvinsky-Lam-Smith formulas, confirm the third family of admissible triples proposed by Yuzvinsky in 1984, improve the two infinite families of solutions obtained recently by Lenzhen, Morier-Genoud and Ovsienko, and construct several new infinite families of solutions.