Associators and commutators in alternative algebras
Algebra i logika, Tome 58 (2019) no. 4, pp. 479-485
It is proved that in a unital alternative algebra $A$ of characteristic $\neq 2$, the associator $(a,b,c)$ and the Kleinfeld function $f(a,b,c,d)$ never assume the value $1$ for any elements $a,b,c,d\in A$. Moreover, if $A$ is nonassociative, then no commutator $[a,b]$ can be equal to $1$. As a consequence, there do not exist algebraically closed alternative algebras. The restriction on the characteristic is essential, as exemplified by the Cayley–Dickson algebra over a field of characteristic $2$.
Keywords:
alternative algebra, commutator, Kleinfeld function.
Mots-clés : associator
Mots-clés : associator
@article{AL_2019_58_4_a3,
author = {E. Kleinfeld and I. P. Shestakov},
title = {Associators and commutators in alternative algebras},
journal = {Algebra i logika},
pages = {479--485},
year = {2019},
volume = {58},
number = {4},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/AL_2019_58_4_a3/}
}
E. Kleinfeld; I. P. Shestakov. Associators and commutators in alternative algebras. Algebra i logika, Tome 58 (2019) no. 4, pp. 479-485. http://geodesic.mathdoc.fr/item/AL_2019_58_4_a3/
[1] E. Kleinfeld, “Simple alternative rings”, Ann. Math. (2), 58:3, 544–547 | DOI | MR | Zbl
[2] K. A. Zhevlakov, A. M. Slinko, I. P. Shestakov, A. I. Shirshov, Koltsa, blizkie k assotsiativnym, Nauka, M., 1978 | MR
[3] I. P. Shestakov, “Tsentry alternativnykh algebr”, Algebra i logika, 15:3 (1976), 343—362 | MR
[4] L. Makar-Limanov, “Algebraically closed skew fields”, J. Algebra, 93:1 (1985), 117–135 | DOI | MR | Zbl