Identities of finite-dimensional nilpotent algebras (Q1803024)

The author proves the following main result: Let \(R\) be an \(n\)- dimensional nilpotent algebra (over a field) and \(\dim R^ 2/R^ 3 = 2\). Then \(\dim R^ i/R^{i + 1} \leq 2\), \(i = 2,3,\dots\) and \(R\) satisfies the standard identity \(S_ k(x_ 1,\dots,x_ k) = 0\) where \(k = [(1 + \sqrt{1 + 8n})/2]\). He uses this result to prove that \(S_ k = 0\) is the minimal identity of a variety which is generated by all \(n\)- dimensional nilpotent algebras in the following cases: \(n \leq 12\), \(n = 15\).