restricted Lie algebra

Definition 1.

Given a modular Lie algebra $L$, that is, a Lie algebra defined over a field ofpositive characteristic $p$, then we say $L$ is restricted if thereexists a power map $\mathrm{[}p\mathrm{]}\mathrm{:}L\mathrm{\to }L$ satisfying

1. ${(\mathrm{ad}a)}^p=\mathrm{ad}(a^{[p]})$ for all $a\in L$ (where ${(\mathrm{ad}a)}^p=\mathrm{ad}a\mathrm{\cdots }\mathrm{ad}a$ in the associative product of linear transformations in $𝔤{𝔩}_{k}V={\mathrm{End}}_{k}V$.)

2. For every $\alpha \in k$ and $a\in L$ then ${(\alpha a)}^{[p]}={\alpha }^p{a}^{[p]}$.

3. For all $a,b\in L$,

   $${(a+b)}^{[p]}=a^{[p]}+b^{[p]}+\sum _{i=1}^{p-1}{s}_i(a,b)$$

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

   $${(\mathrm{ad}(a\otimes x+b\otimes 1))}^{p-1}(a\otimes 1)=\sum _{i=1}^{p-1}i{s}_i(a,b)\otimes x^{i-1}$$

   in $L{\otimes }_{k}k[x]$.

The definition and terminology of a restricted Lie algebra was developed byN. Jacobson as a method to mimic the properties of Lie algebras of characteristic 0. The usual methods of using minimal polynomials to establishthe Jordan normal form of a transform fail in positive characteristic ascertain polynomials become inseparable or reducible. Thus one cannotsimply establish the typical nilpotent+semisimple decomposition of elements.

However, given a linear Lie algebra (subalgebra of $𝔤{𝔩}_{k}V$) over a field $k$ of positive characteristic $p$ then the map $x\mapsto x^p$ on the matrices captures many of the properties of the field. For example, a diagonalmatrix $D$ with entries in ${\mathbb{Z}}_p$ satisfies $D^p=D$ simply becausethe power map $p$ is a field automorphism. Thus in various ways the powermap captures the requirements of semisimple and toral elements of Liealgebras. By modifying the definitions of semisimple, toral, and nilpotentelements to use the power maps of the field, Jacobson and others were able toreproduce much of the classical theory of Lie algebras for restricted Lie algebras.

The definition given above reflects 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 $L^p\subseteq L$.

Definition 2.

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

Remark 3.

It is generally not true that a restrictable Lie algebra has a uniquepower mapping. Notice that the definition of a power mapping relatesthe power mapping to the linear power mapping of the adjoint representation.This suggests (correctly) that power maps can be defined in various waysbut 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 representations, Monographs and Textbooks in Pure and Applied Mathematics,vol. 116, Marcel Dekker Inc., New York, 1988.