restricted Lie algebra

Definition 1.

Given a modular Lie algebra L, that is, a Lie algebraMathworldPlanetmath defined over a field of positive characteristic p, then we say L is restricted if there exists a power map [p]:LL satisfying

  1. 1.

    (ada)p=ad(a[p]) for all aL (where (ada)p=adaada in the associative product of linear transformations in 𝔤𝔩kV=EndkV.)

  2. 2.

    For every αk and aL then (αa)[p]=αpa[p].

  3. 3.

    For all a,bL,


    where the terms si(a,b) are determined by the formula


    in Lkk[x].

The definition and terminology of a restricted Lie algebra was developed by N. Jacobson as a method to mimic the properties of Lie algebras of characteristicPlanetmathPlanetmath 0. The usual methods of using minimal polynomialsPlanetmathPlanetmath to establish the Jordan normal form of a transform fail in positive characteristic as certain polynomialsPlanetmathPlanetmath become inseparable or reducible. Thus one cannot simply establish the typical nilpotent+semisimple decomposition of elements.

However, given a linear Lie algebra (subalgebraMathworldPlanetmath of 𝔤𝔩kV) over a field k of positive characteristic p then the map xxp on the matrices captures many of the properties of the field. For example, a diagonal matrixMathworldPlanetmath D with entries in p satisfies Dp=D simply because the power map p is a field automorphism. Thus in various ways the power map captures the requirements of semisimplePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath and toral elements of Lie algebras. By modifying the definitions of semisimple, toral, and nilpotent elementsMathworldPlanetmath to use the power maps of the field, Jacobson and others were able to reproduce much of the classical theory of Lie algebras for restricted Lie algebras.

The definition given above reflectsPlanetmathPlanetmath the abstract requirements for a restricted Lie algebra. However an important observation is that given a linear Lie algebra L over a field of characteristic p, then the usual associative power map serves as a power map for establishing a restricted Lie algebra. The only added requirement is that LpL.

Definition 2.

A Lie algebra is said to be restrictable if it can be given a power mapping which makes it a restricted Lie algebra.

Remark 3.

It is generally not true that a restrictable Lie algebra has a unique power mapping. Notice that the definition of a power mapping relates the power mapping to the linear power mapping of the adjoint representationMathworldPlanetmath. This suggests (correctly) that power maps can be defined in various ways but agree modulo the center of the Lie algebra.

Jacobson, Nathan Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962.

Strade, Helmut and Farnsteiner, Rolf Modular Lie algebras and their representationsPlanetmathPlanetmath, Monographs and Textbooks in Pure and Applied Mathematics, vol. 116, Marcel Dekker Inc., New York, 1988.

Title restricted Lie algebra
Canonical name RestrictedLieAlgebra
Date of creation 2013-03-22 16:26:48
Last modified on 2013-03-22 16:26:48
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 7
Author Algeboy (12884)
Entry type Definition
Classification msc 17B99
Related topic ModularTheory
Defines restricted Lie algebra
Defines modular Lie algebra
Defines restrictable Lie algebra