Dimensionally nilpotent genetic algebras (Q1372976)

Let \(K\) be a field and \(A\) a finite dimensional algebra, then \(A\) is said to be dimensionally nilpotent if there exists a \(K\)-derivation \(d\) such that \(d^{n+1}=0\) but \(d^{n}\neq 0\). The structure of dimensionally nilpotent Lie algebras was studied by G. F. Leger and P. L. Manley for characteristic zero and by J. M. Osborn for characteristic \(p>5\). Moreover, J. M. Osborn studied dimensionally nilpotent commutative and noncommutative Jordan algebras. In this paper the authors prove first that any finite dimensional and dimensionally nilpotent Bernstein algebra over a field of characteristic not two is in fact a genetic algebra. Then they prove that the polyploid gametic algebras \(G(2,2m)\) (\(m\geq 1\)) with two alleles are dimensionally nilpotent. Next they study polyploid gametic algebras with \(n+1\) alleles. They relate the dimensionally nilpotent character of the \(2m\)-ploid algebra with the \(2m-2\)-ploid one. The authors prove that the gametic algebra \(G(n+1,2m)\) is dimensionally nilpotent if and only if \(m=1\) or \(n=1\). In the last sections of the paper under review, the authors study dimensionally nilpotent algebras over commutative rings (with unit). They prove that the dimensional nilpotence is inherited under scalar extension in certain particular cases. They extend some of the results in previous sections to the case of general gametic algebras. They finish the work with a study of the relationship between the dimensionally nilpotent character of an algebra and the one of certain algebras constructed from the first one.