A note on the tensor product of Lie soluble algebras. (Q2642245)

Let \(R\) be a Lie soluble associative algebra and \(dl_L(R)\) be the Lie derived length of \(R\). In this paper it is established that if \(A\) is a Lie metabelian and \(B\) is a Lie nilpotent algebra of class \(c\geq 2\), then \(A\otimes B\) is Lie soluble and \(dl_L(A\otimes B)\leq c+1\). The last result holds also if \(A\) and \(B\) are Lie nilpotent of class at most \(c\geq 2\). If \(A\) and \(B\) are Lie metabelian, then \(dl_L(A\otimes B)\leq 4\). The indicated bounds for \(dl_L(A\otimes B)\) improve results of \textit{D. M. Riley} [Arch. Math. 66, No. 5, 372-377 (1996; Zbl 0854.16023)]. Large classes of strongly Lie soluble and strongly Lie nilpotent algebras are introduced. It is shown that (i) if \(A\) and \(B\) are strongly Lie soluble, then \(A\otimes B\) is strongly Lie soluble and a bound for \(dl_L(A\otimes B)\) is obtained and (ii) if \(A\) and \(B\) are strongly Lie nilpotent, then \(A\otimes B\) is strongly Lie nilpotent and a bound for the Lie nilpotent class of \(A\otimes B\) is obtained.