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Restricted Lie Algebras and Their Envelopes

Published online by Cambridge University Press:  20 November 2018

D. M. Riley  and
A. Shalev
D. M. Riley
Affiliation:
Mathematical Institute University of Oxford 24-29 St. Giles' Oxford OX1 3LB England
A. Shalev
Affiliation:
Department of Mathematics The Hebrew University Jerusalem 91904 Israel
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Abstract

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Let L be a restricted Lie algebra over a field of characteristic p. Denote by u(L) its restricted enveloping algebra and by ωu(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 Dn(L) - L∩ωu(L)n for each n ≥ 1. Using this expression we describe the nilpotence index of ωU(L). We also give a precise characterisation of those L for which ωu(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).

Keywords

16S3017B3517B5016U60
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 1 , 01 February 1995 , pp. 146 - 164
Copyright
Copyright © Canadian Mathematical Society 1995

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