Nilpotent linear transformations and the solvability of power-associative nilalgebras
In 1948, \textit{A. A. Albert} [Trans. Am. Math. Soc. 64, 552--593 (1948; Zbl 0033.15402)] conjectured that the finite-dimensional commutative power-associative nilalgebras over a field are nilpotent. Suttles [Notices Am. Math. Soc. 19, A-566 (1972)] gave a counterexample of dimension \(5\) which is, nevertheless, solvable. Albert's problem was then restated as follows: Is every finite dimensional commutative power-associative nilalgebra over a field solvable? The paper under review is devoted to solve in the affirmative this question for dimension \(7\), and for dimension \(n\) and nilindex \(n-1\) or \(n-2\), provided that the characteristic of the ground field is zero or sufficiently large. Results by \textit{M. Gerstenhaber} [Duke Math. J. 27, 21-31 (1960; Zbl 0213.04101)] are instrumental in the proofs.
commutative power-associative nilalgebra
scientific article; zbMATH DE number 2136670