zbMATH — the first resource for mathematics

Autour des algèbres de Bernstein. (About Bernstein algebras). (French) Zbl 0648.17011
It is proved that a Bernstein K-algebra $$A=Ke\oplus U\oplus V$$ is a Jordan algebra if and only if V $$2=(0)$$ and $$v(vU)=(0)$$ for all v in V. It is studied the orthogonality notion of a Bernstein algebra by proving that if the algebra is not orthogonal, then the dimensions of the subspaces U and V are respectively greater than two and one. It is shown an example of a Bernstein algebra that is not orthogonal of type (4,2).
Reviewer: C.Burgueño

MSC:
 17D92 Genetic algebras 17A30 Nonassociative algebras satisfying other identities 17C99 Jordan algebras (algebras, triples and pairs)
Full Text:
References:
 [1] M. T.Alcalde, C.Burgueño, A.Labra et A.Micali, Sur les algèbres de Bernstein. London Math. Soc., sous presse. · Zbl 0672.17010 [2] P. Holgate, Genetic algebras satisfying Bernstein’s stationary principle. J. London Math. Soc. (2)9, 613-623 (1975). · Zbl 0365.92025 · doi:10.1112/jlms/s2-9.4.613 [3] A.Wörz-Busekros, Algebras in genetics. Lecture Notes in Biomath.36, Berlin-Heidelberg-New York 1980. · Zbl 0431.92017 [4] A.Wörz-Busekros, Further remarks on Bernstein algebras. London Math. Soc., sous presse. · Zbl 0631.17001 [5] A. Wörz-Busekros, Bernstein algebras. Arch. Math.48, 388-398 (1987). · Zbl 0597.17014 · doi:10.1007/BF01189631
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.