RESTRICTED LIE-ALGEBRAS AND THEIR ENVELOPES

Let L be a restricted Lie algebra over a field of characteristic p. Denote by u(L) its restricted enveloping algebra and by omega u(L) the augmentation ideal of u(L). We give an explicit description for the dimension subalgebras of L, namely those ideals of L defined by D-n(L) = L boolean AND omega u(L)(n) for each n greater than or equal to 1. Using this expression we describe the nilpotence index of omega u(L). We also give a precise characterisation of those L for which omega u(L) is a residually nilpotent ideal. In this case we show that the minimal number of elements required to generate an arbitrary ideal of u(L) is finitely bounded if and only if L contains a 1-generated restricted subalgebra of finite codimension. Subsequently we examine certain analogous aspects of the Lie structure of u(L). In particular we characterise L for which u(L) is residually nilpotent when considered as a Lie algebra, and give a formula for the Lie nilpotence index of u(L). This formula is then used to describe the nilpotence class of the group of units of u(L).