{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,16]],"date-time":"2026-02-16T09:33:45Z","timestamp":1771234425699,"version":"3.50.1"},"reference-count":25,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T00:00:00Z","timestamp":1771027200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Colour algebras are noncommutative Jordan algebras and closely related to octonion algebras. They initially emerged in physics since the multiplication of a colour algebra over the real or complex numbers describes the colour symmetry of the Gell\u2013Mann quark model. Over fields of characteristic not equal to two, their structure is now well-known. We initiate the study of colour algebras over a unital commutative base ring R where two is an invertible element, and show when colour algebras can be constructed canonically by employing nondegenerate ternary hermitian forms with trivial determinant. We investigate their structure, their automorphism group and their derivations. We show that there is again a close connection between the colour algebras obtained from hermitian forms and certain types of octonion algebras.<\/jats:p>","DOI":"10.3390\/axioms15020139","type":"journal-article","created":{"date-parts":[[2026,2,16]],"date-time":"2026-02-16T08:38:39Z","timestamp":1771231119000},"page":"139","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Colour Algebras over Rings"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6566-4666","authenticated-orcid":false,"given":"Susanne","family":"Pumpl\u00fcn","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK"}]}],"member":"1968","published-online":{"date-parts":[[2026,2,14]]},"reference":[{"key":"ref_1","first-page":"395","article-title":"On the colour algebra","volume":"5","author":"Elduque","year":"1988","journal-title":"Algebras Groups Geom."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"234","DOI":"10.1016\/0021-8693(92)90014-D","article-title":"Color algebras and affine connections on S6","volume":"149","author":"Elduque","year":"1992","journal-title":"J. Algebra"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1287","DOI":"10.1017\/S0308210500030511","article-title":"Colour algebras and Cayley-Dickson algebras","volume":"125","author":"Elduque","year":"1995","journal-title":"Proc. R. Soc. Edinb. Sect. A Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1006\/jabr.1993.1180","article-title":"A generalization of the algebra of colour I","volume":"160","author":"Schafer","year":"1993","journal-title":"J. Algebra"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"296","DOI":"10.1006\/jabr.1994.1152","article-title":"A generalization of the algebra of colour II","volume":"166","author":"Schafer","year":"1994","journal-title":"J. Algebra"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1006\/jabr.1994.1278","article-title":"Simple noncommutative Jordan algebras satisfying ([x,y],y,y) = 0","volume":"169","author":"Schafer","year":"1994","journal-title":"J. Algebra"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1090\/S0002-9947-1993-1108613-X","article-title":"Composition algebras over algebraic curves of genus 0","volume":"337","author":"Petersson","year":"1993","journal-title":"Trans. Amer. Math. Soc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1080\/00029890.1982.11995467","article-title":"A little color in abstract algebra","volume":"89","author":"Wene","year":"1982","journal-title":"Am. Math. Mon."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2400322","DOI":"10.1002\/andp.202400322","article-title":"An Algebraic Roadmap of Particle Theories Part I: General construction","volume":"537","author":"Furey","year":"2025","journal-title":"Ann. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1477","DOI":"10.1063\/1.523815","article-title":"The algebra of colour","volume":"19","author":"Domokos","year":"1978","journal-title":"J. Math. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"85","DOI":"10.2140\/pjm.1985.116.85","article-title":"Nonassociative algebras with scalar involution","volume":"116","author":"McCrimmon","year":"1985","journal-title":"Pacific J. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"5119","DOI":"10.1080\/00927879508825523","article-title":"Cayley algebra bundles on AK2 revisited","volume":"23","author":"Thakur","year":"1995","journal-title":"Comm. Algebra"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1007\/BF02621604","article-title":"Reduced models of Albert algebras","volume":"223","author":"Petersson","year":"1996","journal-title":"Math. Z."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1007\/s10474-008-6157-3","article-title":"On flexible quadratic algebras","volume":"119","year":"2008","journal-title":"Acta Math. Hungar."},{"key":"ref_15","first-page":"45","article-title":"On compositions and triality","volume":"457","author":"Sridharan","year":"1994","journal-title":"J. Reine Angew. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1016\/j.jalgebra.2025.01.021","article-title":"Octonion algebras over schemes and the equivalence of isotopes and isometric quadratic forms","volume":"669","author":"Hildebrandsson","year":"2025","journal-title":"J. Algebra"},{"key":"ref_17","unstructured":"Achhammer, G. (1995). Albert Algebren \u00dcber Lokal Geringten R\u00e4umen. [Ph.D. Thesis, FernUniversit\u00e4t Hagen]."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Achhammer, G. (1993). The first Tits construction of Albert algebras over locally ringed spaces. Nonassociative Algebra and Its Applications, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-011-0990-1_2"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"582","DOI":"10.1006\/jabr.1997.7161","article-title":"Jordan algebras and F4 bundles over the affine plane","volume":"198","author":"Parimala","year":"1997","journal-title":"J. Algebra"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1023\/A:1001507928187","article-title":"Tits\u2019 constructions of Jordan algebras and F4 bundles on the plane","volume":"119","author":"Parimala","year":"1999","journal-title":"Compos. Math."},{"key":"ref_21","first-page":"361","article-title":"Structurable algebras over affine space","volume":"254","year":"2012","journal-title":"Pacific J. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"4178","DOI":"10.1016\/j.jalgebra.2008.09.011","article-title":"Albert algebras over curves of genus zero and one","volume":"320","year":"2008","journal-title":"J. Algebra"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1142\/S1005386705000076","article-title":"Quaternion algebras over curves of genus 1 without rational points","volume":"12","year":"2005","journal-title":"Algebra Colloq."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"4357","DOI":"10.1080\/00927879808826415","article-title":"Quaternion algebras over elliptic curves","volume":"26","year":"1998","journal-title":"Comm. Alg."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"303","DOI":"10.4153\/CMB-2012-044-7","article-title":"Octonion algebras over rings are not determined by their norms","volume":"57","author":"Gille","year":"2014","journal-title":"Can. Math. 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