restricted Lie algebra

restricted Lie algebra

Definition 1

.

Given a

modular Lie algebra

L

, that is, a

Lie algebra

defined over a field of
positive characteristic

p

, then we say

L

is

restricted

if there
exists a

power map

[

p

]

:

L

→

L

satisfying

1.

(

ad

⁡

a

)

p

=

ad

⁡

(

a

[

p

]

)

for all

a

∈

L

(where

(

ad

⁡

a

)

p

=

ad

⁡

a

⁢

⋯

⁢

ad

⁡

a

in the

associative

product of

linear transformations

in

𝔤

⁢

𝔩

k

⁢

V

=

End

k

⁡

V

.)

2.

For every

α

∈

k

and

a

∈

L

then

(

α

⁢

a

)

[

p

]

=

α

p

⁢

a

[

p

]

.

3.

For all

a

,

b

∈

L

,

(

a

+

b

)

[

p

]

=

a

[

p

]

+

b

[

p

]

+

∑

i

=

1

p

-

1

s

i

⁢

(

a

,

b

)

where the terms

s

i

⁢

(

a

,

b

)

are determined by the formula

(

ad

⁡

(

a

⊗

x

+

b

⊗

1

)

)

p

-

1

⁢

(

a

⊗

1

)

=

∑

i

=

1

p

-

1

i

⁢

s

i

⁢

(

a

,

b

)

⊗

x

i

-

1

in

L

⊗

k

k

⁢

[

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

characteristic

0. The usual methods of using

minimal polynomials

to establish
the Jordan normal form of a transform fail in positive characteristic as
certain

polynomials

become inseparable or

reducible

. Thus one cannot
simply 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

↦

x

p

on the matrices captures many of the properties of the field. For example, a

diagonal
matrix

D

with entries in

ℤ

p

satisfies

D

p

=

D

simply because
the power map

p

is a

field automorphism

. Thus in various ways the power
map captures the requirements of

semisimple

and toral elements of Lie
algebras. By modifying the definitions of semisimple, toral, and

nilpotent
elements

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

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

⊆

L

.

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 representation

.
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

representations

, 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

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