Nilpotency of commutative finitely generated algebras satisfying \(L_x^3+\gamma\, L_{x^3}\), \(\gamma =1,0\)
This paper proves the following variation of Albert's conjecture: ``Every commutative non-associative finitely-generated algebra over a field of characteristic other than \(2\) and \(3\), satisfying one of the identities \[ \text{(i)} \quad L_x^3+L_{(x^2)x}=0,\quad \text{(ii)} \quad L_x^3=0 \] is nilpotent (here \(L_x\) stands for the left multiplication operator \(z\mapsto xz\)). Albert's original conjecture was that every commutative finite-dimensional power-associative nilalgebra is nilpotent. D. A. Suttles gave an example in 1972 of a commutative power-associative nilalgebra of dimension \(5\) which was not nilpotent (but it is solvable). Thus as explained by the author in the paper under review, the conjecture was modified to `` every commutative finite-dimensional power associative nilalgebra is solvable''. This modified conjecture is still open nowadays (though solved for dimensions in the range \(1,\ldots, 8\)). The authors explain in the paper that some partial advances have been given in dimensions \(9, 10\) or when the nilindex is close to the dimension (in this case the conjecture has been solved).
scientific article; zbMATH DE number 5931634
scientific article