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$ \mathbb{Z}_2^n$-graded quasialgebras and the Hurwitz problem on compositions of quadratic forms


Authors: Ya-Qing Hu, Hua-Lin Huang and Chi Zhang
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 16S35, 16W50, 11E25
DOI: https://doi.org/10.1090/tran/6946
Published electronically: July 7, 2017
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Abstract: We introduce a series of $ \mathbb{Z}_2^n$-graded quasialgebras $ \mathbb{P}_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.


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Additional Information

Ya-Qing Hu
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405 – and – School of Mathematics, Shandong University, Jinan 250100, People’s Republic of China
Email: yaqhu@indiana.edu, yachinghu@mail.sdu.edu.cn

Hua-Lin Huang
Affiliation: School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, People’s Republic of China – and – School of Mathematics, Shandong University, Jinan 250100, People’s Republic of China
Email: hualin.huang@hqu.edu.cn, hualin@sdu.edu.cn

Chi Zhang
Affiliation: School of Mathematics, Shandong University, Jinan 250100, People’s Republic of China
Email: chizhang@mail.sdu.edu.cn

DOI: https://doi.org/10.1090/tran/6946
Keywords: Quasialgebra, Hurwitz problem, square identity
Received by editor(s): June 4, 2015
Received by editor(s) in revised form: March 14, 2016
Published electronically: July 7, 2017
Additional Notes: This research was supported by SRFDP 20130131110001, SDNSF ZR2013AM022, NSFC 11471186 and 11571199.
Article copyright: © Copyright 2017 American Mathematical Society