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| Format: | Preprint |
| Published: |
2022
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| Online Access: | https://arxiv.org/abs/2211.01073 |
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| _version_ | 1866911759942549504 |
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| author | Fox, Daniel J. F. |
| author_facet | Fox, Daniel J. F. |
| contents | The sectional nonassociativity of a metrized (not necessarily associative or unital) algebra is defined analogously to the sectional curvature of a pseudo-Riemannian metric, with the associator in place of the Levi-Civita covariant derivative. For commutative real algebras nonnegative sectional nonassociativity is usually called the Norton inequality, while a sharp upper bound on the sectional nonassociativity of the Jordan algebra of Hermitian matrices over a real Hurwitz algebra is closely related to the Böttcher-Wenzel-Chern-do Carmo-Kobayashi inequality. These and other basic examples are explained, and there are described some consequences of bounds on sectional nonassociativity for commutative algebras. A technical point of interest is that the results work over the octonions as well as the associative Hurwitz algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_01073 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Sectional nonassociativity of metrized algebras Fox, Daniel J. F. Rings and Algebras The sectional nonassociativity of a metrized (not necessarily associative or unital) algebra is defined analogously to the sectional curvature of a pseudo-Riemannian metric, with the associator in place of the Levi-Civita covariant derivative. For commutative real algebras nonnegative sectional nonassociativity is usually called the Norton inequality, while a sharp upper bound on the sectional nonassociativity of the Jordan algebra of Hermitian matrices over a real Hurwitz algebra is closely related to the Böttcher-Wenzel-Chern-do Carmo-Kobayashi inequality. These and other basic examples are explained, and there are described some consequences of bounds on sectional nonassociativity for commutative algebras. A technical point of interest is that the results work over the octonions as well as the associative Hurwitz algebras. |
| title | Sectional nonassociativity of metrized algebras |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2211.01073 |