modular theory

modular theory

A

representation

of a group/

algebra

into

G

⁢

L

⁢

(

V

,

k

)

,

E

⁢

n

⁢

d

k

⁢

V

,

𝔤

⁢

𝔩

k

⁢

(

V

)

, etc. is called

modular

if the

characteristic

of

k

is

positive

. An algebra

A

is called

modular

if the characteristic of the field

k

is positive (thus prime for some

p

.) As the algebra can be represented
in

E

⁢

n

⁢

d

k

⁢

A

by left or right

multiplication

(often called the

regular

or

adjoint representation

) this definition
matches the requirements for a

modular representation

.

Remark 1

.

Modular representations and modular theory in this sense should not be confused with

modular forms

from

number theory

. Though applications of one to another may exist, these two topics are generally unrelated.

Typically theory for representations over fields of characteristic 0 cannot be
transplanted directly to modular representations. For example, Maschke’s

theorem

, Lie’s theorem, the use of the

Killing form

for a

Lie algebra

all fail in various ways in modular representations. There are two common obstructions in modular representations.

1.

A theorem/

formula

in characteristic 0 makes use of

rational

fractions

a

/

b

where

p

|

b

. Thus the fraction causes

division

by 0 over characteristic

p

. Examples of this obstruction include Maschke’s theorem and the definition
of exponentials of

nilpotent matrices

.

2.

A theorem/formula involves

polynomials

which are inseparable, or have
repeated roots over fields with positive characteristic. This is especially
problematic when the polynomials are the

minimal

or

characteristic polynomials

of

linear transformations

for this causes the

rational canonical form

and

Jordan canonical form

of the transforms to change. Examples of this problem include Lie’s theorem.

There are also two common workarounds for these obstructions.

1.

Treat the rational coefficients as formal coefficients. For example,
start in characteristic 0 and define an

integer

subalgebra

with basis elements of the form

x

n

n

!

(or similarly useful

combinations

.) Then tensor (over

ℤ

) the

ℤ

-subalgebra with

k

. Thus the fractions
of the form

1

/

n

!

are not actually coefficients in

k

but formal coefficients and so they do not cause division by 0. This technique is used in the theory
of

Chevalley groups

over arbitrary fields as it allows for version of the
exponential of a nilpotent element of a Lie algebra.

2.

Make

restrictions

on the

dimensions

of the representation, for

instance

,

(

p

,

dim

⁡

V

)

=

1

or

x

n

=

0

for

n

<

p

etc. Thus the division by 0 and polynomial oddities are avoided. For example, with Maschke’s theorem the

solution

is to
assume that the characteristic

p

does not divide the order of the group

G

which is being represented. The second approach to insist on large

p

often
excludes

p

=

2

,

3

but provides workable results for

p

≥

5

, or 7, etc. Then the small prime cases are studied as exceptional examples.

There is a third problem which can arise for modular representations which has no obvious work around. This is when the definition and/or theorems still work but their

implications

are useless.

For example, with the Killing form of a Lie algebra we take the trace of a linear transformation. However, over characteristic

p

it is possible for the

identity matrix

to have trace 0, for example, in dimension

d

=

m

⁢

p

for any integer

m

. This situation cannot be be avoided as easily as with Maschke’s theorem by assuming restrictions on the dimension of the representation. For we can use

d

=

2

and the

diagonal matrix

D

⁢

i

⁢

a

⁢

g

⁢

(

1

,

p

-

1

)

to obtain a

similar

problem with the trace. And this problem embeds into all higher dimensions thus a straight forward use of trace would work only in 1 dimensional representations.

There is no problem with the definition of the Killing form for modular representations but the results may no longer be applicable. In these situations usually entirely new approaches are required.

Title

modular theory

Canonical name

ModularTheory

Date of creation

2013-03-22 16:26:45

Last modified on

2013-03-22 16:26:45

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

6

Author

Algeboy (12884)

Entry type

Definition

Classification

msc 17B99

Related topic

RestrictedLieAlgebra

Defines

modular theory

Defines

modular representation

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