{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:02:25Z","timestamp":1760144545035,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2024,4,29]],"date-time":"2024-04-29T00:00:00Z","timestamp":1714348800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Centre de Recherches Math\u00e9matiques","award":["RGPIN-2020-05020"],"award-info":[{"award-number":["RGPIN-2020-05020"]}]},{"name":"Monica Nevins\u2019 NSERC Discovery","award":["RGPIN-2020-05020"],"award-info":[{"award-number":["RGPIN-2020-05020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.3390\/axioms13050299","type":"journal-article","created":{"date-parts":[[2024,4,30]],"date-time":"2024-04-30T04:01:52Z","timestamp":1714449712000},"page":"299","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Generalization of the First Tits Construction"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-3379-7713","authenticated-orcid":false,"given":"Thomas","family":"Moran","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Avenue, Ottawa, ON K1N 7N5, Canada"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6566-4666","authenticated-orcid":false,"given":"Susanne","family":"Pumpluen","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK"}]}],"member":"1968","published-online":{"date-parts":[[2024,4,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Knus, M.A., Merkurjev, A., Rost, M., and Tignol, J.-P. (1998). The Book of Involutions, AMS Colloquium Publications.","DOI":"10.1090\/coll\/044"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1007\/s11856-021-2175-4","article-title":"On R-triviality of F4","volume":"244","author":"Thakur","year":"2021","journal-title":"Isr. J. Math."},{"key":"ref_3","first-page":"497","article-title":"On R-triviality of F4, II","volume":"14","author":"Thakur","year":"2021","journal-title":"M\u00fcnster J. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"4701","DOI":"10.1090\/tran\/7850","article-title":"Automorphisms of Albert algebras and a conjecture of Tits and Weiss II","volume":"372","author":"Thakur","year":"2019","journal-title":"Trans. Amer. Math. Soc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1007\/s11856-021-2091-7","article-title":"The cyclicity problem for Albert algebras","volume":"241","author":"Thakur","year":"2021","journal-title":"Isr. J. Math."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Garibaldi, S., Petersson, H.P., and Racine, M.L. (2023). Albert algebras over Z and other rings. Forum. Math. Sigma, 11.","DOI":"10.1017\/fms.2023.7"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"239","DOI":"10.2307\/2372761","article-title":"Autotopism groups of some finite nonassociative algebras","volume":"84","author":"Sandler","year":"1962","journal-title":"AMS J. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2401","DOI":"10.1016\/j.jpaa.2018.08.018","article-title":"Nonassociative cyclic extensions of fields and central simple algebras","volume":"223","author":"Brown","year":"2019","journal-title":"J. Pure Appl. Algebra"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"106540","DOI":"10.1016\/j.jpaa.2020.106540","article-title":"The automorphisms of generalized cyclic Azumaya algebras","volume":"225","year":"2021","journal-title":"J. Pure Appl. Algebra"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1007\/s11856-014-0021-7","article-title":"Nonassociative cyclic algebras","volume":"200","author":"Steele","year":"2014","journal-title":"Isr. J. Math."},{"key":"ref_11","unstructured":"Steele, A. (2013). Some New Classes of Algebras. [Ph.D. Thesis, University of Nottingham]. Available online: http:\/\/eprints.nottingham.ac.uk\/13934\/1\/PhdthesisFinal.pdf."},{"key":"ref_12","first-page":"609","article-title":"Space-time block codes from nonassociative division algebras","volume":"5","author":"Unger","year":"2011","journal-title":"Adv. Math. Commun."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Steele, A., Pumpl\u00fcn, S., and Oggier, F. (2012, January 3\u20137). MIDO space-time codes from associative and non-associative cyclic algebras. Proceedings of the 2012 IEEE Information Theory Workshop, Lausanne, Switzerland.","DOI":"10.1109\/ITW.2012.6404655"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/0001-8708(70)90019-8","article-title":"Forms permitting composition","volume":"4","author":"Schafer","year":"1970","journal-title":"Adv. Math."},{"key":"ref_15","unstructured":"McCrimmon, K. (2004). A Taste of Jordan Algebras, Springer."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3041","DOI":"10.1090\/S0002-9947-2012-05710-2","article-title":"Automorphisms of Albert algebras and a conjecture of Tits and Weiss","volume":"365","author":"Thakur","year":"2013","journal-title":"Trans. Amer. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/0021-8693(86)90024-4","article-title":"Jordan algebras of degree 3 and the Tits process","volume":"98","author":"Petersson","year":"1986","journal-title":"J. Algebra"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Jacobson, N. (1968). Structure and Representations of Jordan Algebras, AMS. Providence, AMS. Colloquium Publications.","DOI":"10.1090\/coll\/039"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/5\/299\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:36:31Z","timestamp":1760106991000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/5\/299"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,4,29]]},"references-count":18,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2024,5]]}},"alternative-id":["axioms13050299"],"URL":"https:\/\/doi.org\/10.3390\/axioms13050299","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,4,29]]}}}