Associative algebras with one Engel-type relation (Q1972666)

Let \(A\) be an associative algebra over the field of rationals \(\mathbb{Q}\) defined by generators \(a,x\) and the relation \([[a,x],x]=0\). The author studies various ``cones'' in \(A\), where by the cone generated by a set \(M\) the author means the set of all linear combinations of elements in \(M\) with nonnegative coefficients. In particular, if \(C(x^n)\) denotes the cone generated by all monoms containing \(x^n\) then, in the case \((ax)^N\in C(x^n)\) for some \(N\), the power \(N\) is at least a quadratic function in \(n\).