Power closure and the Engel condition

A Liep-algebraL is calledn-power closed if, in every section ofL, any sum ofp ⁱ⁺ⁿ th powers is ap ⁱ th power (i>0). It is easy to see that ifL isp ⁿ -Engel then it isn-power closed. We establish a partial converse to this statement: ifL is residually nilpotent andn-power closed for somen≥0 thenL is (3p ^{n +2} +1)-Engel ifp>2 and (3 · 2 ⁿ⁺³+1)-Engel ifp=2. In particular, thenL is locally nilpotent by a theorem of Zel’manov. We deduce that a finitely generated pro-p group is a Lie group over thep-adic field if and only if its associated Liep-algebra isn-power closed for somen. We also deduce that any associative algebraR generated by nilpotent elements satisfies an identity of the form (x+y) ^{p n} =x ^{p n} +y ^{p n} for somen≥1 if and only ifR satisfies the Engel condition. This project was supported by the CNR in Italy and NSF-EPSCoR in Alabama during the first author’s stay at the Università di Palermo.