$\mathbb {Z}_2^n$-graded quasialgebras and the Hurwitz problem on compositions of quadratic forms
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- by Ya-Qing Hu, Hua-Lin Huang and Chi Zhang PDF
- Trans. Amer. Math. Soc. 370 (2018), 241-263 Request permission
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.References
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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
- MR Author ID: 694521
- 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
- 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.
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 241-263
- MSC (2010): Primary 16S35, 16W50, 11E25
- DOI: https://doi.org/10.1090/tran/6946
- MathSciNet review: 3717980