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Cubic and symmetric compositions over rings

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Abstract

We consider generalized symmetric compositions over a ring k on the one hand, and unital algebras with multiplicative cubic forms on the other. Given a primitive sixth root of unity in k, we construct functors between these categories which are equivalences if 3 is a unit in k. This extends to arbitrary base rings, and with new proofs, results of Elduque and Myung on non-degenerate symmetric compositions and separable alternative algebras of degree 3 over fields. It also answers a problem posed in “The Book of Involutions” [Knus et al.: American Mathematical Society Colloquium Publications, vol. 44. American Mathematical Society, Providence, RI (1998), 34.26].

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  1. Föhrenweg 10, 6094, Axams, Austria

    Ottmar Loos

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  1. Ottmar Loos
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Correspondence to Ottmar Loos.

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Loos, O. Cubic and symmetric compositions over rings. manuscripta math. 124, 195–236 (2007). https://doi.org/10.1007/s00229-007-0111-5

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  • DOI: https://doi.org/10.1007/s00229-007-0111-5

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