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About the Engel condition in Bernstein algebras. (Autour de la condition d’Engel dans les algèbres de Bernstein.) (French) Zbl 0956.17021
Let $$(A, \omega)$$ be a Bernstein algebra over a field $$K$$ of characteristic different from $$2.$$ There are always nonzero idempotent elements in $$A.$$ Let $$e$$ be one of them and $$U_e$$ and $$V_e$$ be the subspaces defined by $$U_e = \{ x\in \ker \omega : ex= \frac 12 x\}$$ and $$V_e = \{ x\in \ker \omega : ex=0 \} .$$ Then $$A = Ke \oplus U_e \oplus V_e .$$ Moreover $$N = \ker \omega$$ agrees with $$U_e \oplus V_e$$.
The Bernstein algebra $$(A, \omega)$$ satisfies the $$k$$th Engel condition if for every $$x\in N$$ the multiplication operator $$L_x$$ satisfies that the restriction to $$N$$ of $$L_x ^k$$ is the zero map and $$k\geq 1$$ is the lowest positive integer with this property. On the other hand, $$(A, \omega)$$ satisfies the $$k$$th weak Engel condition if $$L_{v|U}^{k} = 0$$ for all $$v\in V_e$$ and $$k\geq 1$$ is the lowest positive integer with this property. This is a condition that does not depend on the idempotent $$e.$$
The authors prove the nilpotence of $$N$$ in each of the following cases: (i) if $$A$$ satisfies the second Engel condition; (ii) if $$A$$ has finite dimension and satisfies the second weak Engel condition (so for Bernstein-Jordan algebras of finite dimension the ideal $$N$$ is nilpotent); (iii) for $$(A, \omega)$$ finite-dimensional satisfying the third Engel condition; (iv) if $$A$$ has dimension $$\leq 5$$ and satisfies some Engel condition.
Moreover they show that neither the third weak Engel condition nor the fourth Engel condition imply nilpotence of $$N.$$
The last section is devoted to obtain some consequences for the structure of Bernstein algebras which are train algebras.
##### MSC:
 17D92 Genetic algebras
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##### References:
 [1] Albert A.A., Trans. Amer. Math. Soc. 69 pp 503– (1950) [2] DOI: 10.1215/S0012-7094-60-02702-2 · Zbl 0213.04101 · doi:10.1215/S0012-7094-60-02702-2 [3] DOI: 10.1080/00927879508825216 · Zbl 0829.17028 · doi:10.1080/00927879508825216 [4] DOI: 10.1007/BF01237566 · Zbl 0643.17018 · doi:10.1007/BF01237566 [5] Koulibaly A., Algebras, Groups and Geometries 8 pp 145– (1991) · Zbl 0795.17039 [6] Lyubich Y. I., Biomathenmatics 22 (1992) [7] Micali A., Bull. Soc. Math. Belgique 45 pp 5– (1993) [8] DOI: 10.1016/0024-3795(93)00159-W · Zbl 0831.17015 · doi:10.1016/0024-3795(93)00159-W [9] DOI: 10.1016/0024-3795(91)90092-B · Zbl 0728.17020 · doi:10.1016/0024-3795(91)90092-B [10] DOI: 10.2307/2372100 · Zbl 0034.02004 · doi:10.2307/2372100 [11] DOI: 10.1017/S0013091500005411 · Zbl 0754.17029 · doi:10.1017/S0013091500005411 [12] Worz-Busekros A., Algebras in Genetics 36 (1980) [13] Zhevlakov K.A., Rings that are nearly associative (1982) · Zbl 0487.17001
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