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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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