A generalization of the first Tits construction

Pumpluen, Susanne; Moran, Thomas

Abstract

Let F be a field of characteristic, not 2 or 3. The first Tits construction is a well-known tripling process to construct separable cubic Jordan algebras, especially Albert algebras. We generalize the first Tits construction by choosing the scalar employed in the tripling process outside of the base field. This yields a new family of non-associative unital algebras which carry a cubic map, and maps that can be viewed as generalized adjoint and generalized trace maps. These maps display properties often similar to the ones in the classical setup. In particular, the cubic norm map permits some kind of weak Jordan composition law.

• Journal Article Type: Article
• Acceptance Date: Apr 24, 2024
• Online Publication Date: Apr 29, 2024
• Publication Date: 2024-05
• Deposit Date: Apr 24, 2024
• Publicly Available Date: Apr 24, 2024
• Journal: Axioms
• Publisher: MDPI
• Peer Reviewed: Peer Reviewed
• Volume: 13
• Issue: 5
• Article Number: 299
• DOI: https://doi.org/10.3390/axioms13050299
• Keywords: non-associative algebras; first Tits construction; Jordan algebras; generalized cubic algebras
• Public URL: https://nottingham-repository.worktribe.com/output/34098709
• Publisher URL: https://www.mdpi.com/journal/axioms