Let K be a commutative field of characteristic zero. Let A be a finite dimensional algebra over K, not necessarily commutative and D(A) its noncommutative duplicate. In this paper, we apply Whitehead’s theorem (or Second Whitehead lemma for alternative algebras, see [11, p. 67] to the noncommutative duplicate of an alternative algebra and a noncommutative Jordan algebra.

duplicate of algebras, noncommutative Jordan algebras, semi-direct product of algebras.

Received: July 2, 2021; Accepted: August 23, 2021; Published: November 15, 2021

How to cite this article: Alexis Tapsoba, Moussa Ouattara and Nakelgbamba Boukary Pilabré, Noncommutative Duplicate of a Jordan Algebra and Whitehead’s Lemma, Far East Journal of Applied Mathematics 111(1) (2021), 31-48. DOI: 10.17654/AM111010031

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] A. H. Boers, Duplication of algebras IV, Indag. Math. 22 (2011), 55-68.
[2] S. Eilenberg, Extensions of general algebras, Ann. Soc. Pol. Math. 21 (1948), 125-134.
[3] I. M. H. Etherington, Duplicate of linear algebras, Proc. Edinb. Math. Soc. 2(6) (1941), 222-230.
[4] N. Jacobson, Structure and representations of Jordan algebras, A.M.S. Colloq. Pub. Vol. 39, Providence, Rhode Island, 1968.
[5] A. Micali, A. Koulibaly and M. Ouattara, Dupliquée d’une algèbre et le Théorème de Whitehead, Indag. Math. (N.S.) 5 (1994), 341-351.
[6] A. Micali and M. Ouattara, Sur la dupliquée d’une algebra, Bull. Soc. Math. Belg. Sér. B 45 (1993), 5-24.
[7] A. Micali and M. Ouattara, Dupliquée d’une algèbre et le Théorème d’Etherington, Linear Algebra Appl. 153 (1991), 193-207.
[8] Albert Nijenhuis, Sur une classe de propriétés communes à quelques types différentes d’algèbres, L’Enseignement Mathmématique 14 (1968).
[9] Moussa Ouattara, Autour de la dupliquée, Indag. Math. (N.S.) 2(1) (1991), 99-104.
[10] Nakelgbamba Boukary Pilabré, Come J. A. Béré and Akry Koulibaly, Noncommutative duplicate and Jordan algebras II, Int. J. Math. Sci. Appl. 1, No. 1, Paper No. 23, 2011, 10 pp.
[11] R. D. Schaffer, An Introduction to Nonassociative Algebras, Academic Press, New York, 1966.